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# Long-time dynamics of the Cahn–Hilliard equation with kinetic rate dependent
dynamic boundary conditions

Harald Garcke 111Fakultät für Mathematik, Universität Regensburg, 93053
Regensburg, Germany<EMAIL_ADDRESS><EMAIL_ADDRESS> Patrik Knopf
11footnotemark: 1 Sema Yayla 222Department of Mathematics, Faculty of Science,
Hacettepe University, Beytepe 06800, Ankara, Turkey
<EMAIL_ADDRESS>
###### Abstract

We consider a Cahn–Hilliard model with kinetic rate dependent dynamic boundary
conditions that was introduced by Knopf, Lam, Liu and Metzger (ESAIM Math.
Model. Numer. Anal., 2021) and will thus be called the KLLM model. In the
aforementioned paper, it was shown that solutions of the KLLM model converge
to solutions of the GMS model proposed by Goldstein, Miranville and Schimperna
(Physica D, 2011) as the kinetic rate tends to infinity. We first collect the
weak well-posedness results for both models and we establish some further
essential properties of the weak solutions. Afterwards, we investigate the
long-time behavior of the KLLM model. We first prove the existence of a global
attractor as well as convergence to a single stationary point. Then, we show
that the global attractor of the GMS model is stable with respect to
perturbations of the kinetic rate. Eventually, we construct exponential
attractors for both models, and we show that the exponential attractor
associated with the GMS model is robust against kinetic rate perturbations.

This is a preprint version of the paper. Please cite as:
H. Garcke, P. Knopf, S. Yayla. Nonlinear Analysis, 215:112619, 2022
https://doi.org/10.1016/j.na.2021.112619

Keywords: Cahn-Hilliard equation, dynamic boundary conditions, long-time
dynamics, stability of global attractors, robustness of exponential
attractors.
2010 MSC classification: 35B40, 35B41, 35K35 35K61 35Q92, 37L30,

##  1 Introduction

The Cahn–Hilliard equation was originally introduced in [7] to describe
spinodal decomposition in binary alloys. Meanwhile, it has become one of the
most popular models to describe various kinds of phase separation phenomena
arising, for instance, in materials science, life sciences and image
processing. The standard Cahn–Hilliard equation as proposed in [7] reads as
follows:

$\displaystyle\partial_{t}u=m_{\Omega}\Delta\mu$ $\displaystyle\quad\text{in
}Q:=\Omega\times(0,\infty),$ (1.1a) $\displaystyle\mu=-\varepsilon\Delta
u+\varepsilon^{-1}F^{\prime}(u)$ $\displaystyle\quad\text{in }Q,$ (1.1b)
$\displaystyle u|_{t=0}=u_{0}$ $\displaystyle\quad\text{in }\Omega.$ (1.1c)

Here, $\Omega\subset\mathbb{R}^{d}$ stands for a bounded domain (usually
$d\in\\{2,3\\}$) with boundary $\Gamma$, $\Delta$ stands for the Laplace
operator acting on $\Omega$, and $m_{\Omega}$ denotes a mobility parameter.
For simplicity, it is assumed to be a non-negative constant, although non-
constant mobilities find a use in some situations (see, e.g., [22]).

In order to describe a binary mixture, the phase-field variable $u$ represents
the difference in volume fractions of both materials. After a short period of
time, the solution $u$ will attain values close to $\pm 1$ in most parts of
the domain $\Omega$. These regions, which correspond to the pure phases of the
materials, are separated by a diffuse interface whose thickness is
proportional to the parameter $\varepsilon>0$ appearing in (1.1b). In most
applications this interface will be very thin and thus, the parameter
$\varepsilon$ is usually chosen to be very small. The time evolution of the
mixture described by $u$ is driven by the chemical potential $\mu$ in the bulk
(i.e., in $\Omega$). It is given as the derivative of the following
Ginzburg–Landau type free energy:

$\displaystyle E_{\text{bulk}}(u)=\int_{\Omega}\frac{\varepsilon}{2}|\nabla
u|^{2}+\frac{1}{\varepsilon}F(u)\,\mathrm{d}x.$ (1.2)

The bulk potential $F$ is usually double-well shaped. Typically, it attains
its minimum at $-1$ and $1$ and has a local maximum in between at $0$. From a
mathematical point of view, $F$ is responsible for the phase separation since
it is energetically favorable for $u$ to attain values close to $\pm 1$. A
physically relevant choice is the _logarithmic potential_

$\displaystyle
F_{\text{log}}(s)=\frac{\vartheta}{2}\big{(}(1+s)\ln(1+s)+(1-s)\ln(1-s)\big{)}-\frac{\vartheta_{c}}{2}s^{2},\quad
s\in(-1,1)$ (1.3)

with constants $0<\vartheta<\vartheta_{c}$. It is often approximated by the
_smooth double-well potential_

$\displaystyle F_{\text{sdw}}(s)=\frac{1}{4}(s^{2}-1)^{2},\quad
s\in\mathbb{R}$ (1.4)

which is easier to handle in terms of mathematical analysis. In this paper, we
will deal with general smooth potentials satisfying certain polynomial growth
conditions such that (1.4) fits into our setting. However, singular potentials
like (1.3) cannot be taken into account in our approach.

To ensure well-posedness of the system (1.1), certain boundary conditions on
$u$ and $\mu$ need to be imposed. The classical choices are the homogeneous
Neumann conditions

$\displaystyle\partial_{\mathbf{n}}u$ $\displaystyle=0$
$\displaystyle\quad\text{on}\;\;\Sigma:=\Gamma\times(0,\infty),$ (1.5)
$\displaystyle\partial_{\mathbf{n}}\mu$ $\displaystyle=0$
$\displaystyle\quad\text{on}\;\;\Sigma.$ (1.6)

The _no-flux condition_ $\eqref{HNC:2}$ implies mass conservation in the bulk,
meaning that (sufficiently regular) solutions satisfy

$\displaystyle\int_{\Omega}u(t)\,\mathrm{d}x=\int_{\Omega}u(0)\,\mathrm{d}x,\quad
t\in[0,\infty).$ (1.7)

Moreover, due to (1.5) and (1.6), we obtain the following energy dissipation
law:

$\displaystyle\frac{d}{dt}E_{\text{bulk}}\big{(}u(t)\big{)}+m_{\Omega}\int_{\Omega}|\nabla\mu(t)|^{2}\,\mathrm{d}x=0,\quad
t\in[0,\infty).$ (1.8)

Furthermore, $\eqref{HNC:1}$ can be regarded as a _contact angle condition_ as
it enforces the diffuse interface separating the regions of pure materials to
intersect the boundary at a perfect right angle. The Cahn–Hilliard equation
(1.1) with the homogeneous Neumann conditions (1.5) and (1.6) is already very
well understood and has been studied from many different viewpoints (see,
e.g., [1, 6, 11, 22, 23]). In particular, we refer to [61, 56, 49, 34, 21] for
the investigation of long-time behavior.

However, in many situations the contact angle condition (1.5) turned out to be
very restrictive as in many applications the contact angle of the interface
will not only deviate from ninety degrees but also change dynamically over the
course of time. In certain situations (e.g., in hydrodynamic applications), it
also turned out to be essential to model short-range interactions between the
mixture of materials and the solid wall of the container more precisely. To
this end physicists (see [26, 27, 40]) proposed that the total free energy
should contain an additional contribution on the surface being also of
Ginzburg–Landau type:

$E_{\text{surf}}(u)=\int_{\Gamma}\frac{\kappa\delta}{2}\left|\nabla_{\Gamma}u\right|^{2}+\frac{1}{\delta}G(u)\,\mathrm{d}\Gamma.$
(1.9)

Here, $\nabla_{\Gamma}$ denotes the surface gradient operator, the constant
$\kappa\geq 0$ acts as a weight for surface diffusion effects and the
parameter $\delta>0$ is related to the thickness of the diffuse interface on
the boundary. Moreover, the function $G$ is an additional surface potential.
If phase separation processes are expected to also occur on the boundary it
makes sense to assume that $G$ exhibits a double-well structure similar to
$F$. In this paper, we will thus impose similar conditions on $G$ as on $F$
such that the choice $G=F_{\mathrm{sdw}}$ is admissible. The total free energy
$E=E_{\text{bulk}}+E_{\text{surf}}$ then reads as

$E(u)=\int_{\Omega}\frac{\varepsilon}{2}|\nabla
u|^{2}+\frac{1}{\varepsilon}F(u)\,\mathrm{d}x+\int_{\Gamma}\frac{\kappa\delta}{2}\left|\nabla_{\Gamma}u\right|^{2}+\frac{1}{\delta}G(u)\,\mathrm{d}\Gamma.$
(1.10)

Associated with this total free energy, various Cahn-Hilliard type systems
with dynamic boundary conditions have been proposed and investigated in the
literature (see,e.g., [16, 18, 31, 32, 19, 45, 48, 51, 52, 55, 60, 28, 59, 30,
8, 9, 17, 50]). In recent times, dynamic boundary conditions which also
exhibit a Cahn–Hilliard type structure have become very popular. Therefore, we
want to highlight the Cahn–Hilliard equation subject to a class of dynamic
boundary conditions of Cahn–Hilliard type depending on a parameter
$L\in[0,\infty]$:

$\displaystyle\partial_{t}u=m_{\Omega}\Delta\mu,$
$\displaystyle\mu=-\varepsilon\Delta u+\varepsilon^{-1}F^{\prime}(u)$
$\displaystyle\quad\text{in }Q,$ (1.11a)
$\displaystyle\partial_{t}v=m_{\Gamma}\Delta_{\Gamma}\theta-\beta
m_{\Omega}\partial_{\bm{n}}\mu,\quad$
$\displaystyle\theta=-\delta\kappa\Delta_{\Gamma}v+\delta^{-1}G^{\prime}(v)+\varepsilon\partial_{\bm{n}}u$
$\displaystyle\quad\text{on }\Sigma,$ (1.11b) $\displaystyle u|_{\Sigma}=v$
$\displaystyle\quad\text{on }\Sigma,$ (1.11c)
$\displaystyle\left\\{\begin{aligned} \mu|_{\Sigma}&=\beta\theta\\\
L\partial_{\bm{n}}\mu|_{\Sigma}&=\beta\theta-\mu|_{\Sigma}\\\
\partial_{\bm{n}}\mu|_{\Sigma}&=0\\\ \end{aligned}\right.$
$\displaystyle\begin{aligned} &\text{if}\;L=0,\\\
&\text{if}\;L\in(0,\infty),\\\ &\text{if}\;L=\infty,\\\ \end{aligned}$
$\displaystyle\quad\text{on }\Sigma,$ (1.11d)
$\displaystyle(u,v)|_{t=0}=(u_{0},v_{0})$
$\displaystyle\text{with}\;u_{0}|_{\Sigma}=v_{0}$ $\displaystyle\quad\text{in
}\Omega\times\Gamma.$ (1.11e)

The chemical potentials $\mu$ and $\theta$ which are coupled by the boundary
condition $\eqref{CH:INT:4}$ describe the chemical interaction between the
materials in the bulk and the materials on the surface. The constant $1/L$ is
related to a _kinetic rate_ , and the term $L\partial_{\mathbf{n}}\mu$
describes adsorption and desorption processes. Here, the mass flux
$-m_{\Omega}\partial_{\mathbf{n}}\mu$ (which describes the motion of the
materials towards and away from the boundary) is directly influenced by
differences in the chemical potentials through the condition (1.11d). As the
models corresponding to the cases $L=0$, $L=\infty$ and $0<L<\infty$ were
introduced separately in different articles in the literature, we are now
going to briefly highlight their most important features.

##### The case $L=0$ (GMS model).

The system (1.11) was first introduced with $L=0$ in (1.11d) by G. Goldstein,
A. Miranville and G. Schimperna [36]. In view of the authors’ initials we will
refer to this system as the GMS model.

It can be regarded as an extension of a model that was previously introduced
by Gal [31] where the equation
$\partial_{t}v=-\beta\partial_{\bm{n}}\mu+\gamma\mu$ on $\Sigma$ (for some
additional constant $\gamma$) was proposed instead of $\eqref{CH:INT:2}$.
Here, the chemical potentials in the bulk and on the boundary can differ only
by the factor $\beta$, i.e., they are directly proportional. In other words,
this means that the chemical potentials $\mu$ and $\theta$ are always in
chemical equilibrium. It is worth mentioning that in [36], $\beta$ is even
allowed to be a uniformly positive function in $L^{\infty}(\Gamma)$. However,
in this paper we restrict ourselves to the case where $\beta$ is a positive
constant.

We observe that any (sufficiently regular) solution to the GMS system
satisfies the mass conservation law

$\displaystyle\beta\int_{\Omega}u(t)\,\mathrm{d}x+\int_{\Gamma}v(t)\,\mathrm{d}\Gamma=\beta\int_{\Omega}u(0)\,\mathrm{d}x+\int_{\Gamma}v(0)\,\mathrm{d}\Gamma,\quad
t\in[0,\infty),$ (1.12)

meaning that the parameter $\beta$ can be interpreted as a weight for the bulk
mass compared to the surface mass. Moreover, the energy dissipation law

$\displaystyle\frac{d}{dt}E\big{(}u(t),v(t)\big{)}+m_{\Omega}\int_{\Omega}|\nabla\mu(t)|^{2}\,\mathrm{d}x+m_{\Gamma}\int_{\Gamma}|\nabla_{\Gamma}\theta(t)|^{2}\,\mathrm{d}\Gamma=0$
(1.13)

is satisfied for all $t\in[0,\infty)$. In particular, we observe that the
dissipation rate is strongly influenced by the values of the mobilities
$m_{\Omega}$ and $m_{\Gamma}$.

The weak well-posedness and some results on long-time behavior were
established in [36]. We will present the well-posedness result as well as some
additional important properties of weak solutions in Section 3.2. A summary of
the results on long-time behavior is given in Section 4.2.

We further point out that numerical analysis for the GMS model as well as some
numerical simulations can be found in [37]. A nonlocal variant of the GMS
model (including a nonlocal dynamic boundary condition) was proposed and
analyzed in [44].

##### The case $L=\infty$ (LW model).

Some time after the introduction of the GMS model, the system (1.11) with
$L=\infty$ in (1.11d) was derived by C. Liu and H. Wu [46] via an energetic
variational approach. We will thus call this system the LW model.

The crucial difference to the GMS model is that the Dirichlet type boundary
condition $\beta\theta=\mu$ is replaced by the no mass flux condition
$\partial_{\mathbf{n}}\mu=0$. This means that the chemical potentials $\mu$
and $\theta$ are not directly coupled. However, mechanical interactions
between the bulk and the surface materials are still taken into account
through the trace condition (1.11c) for the phase-field variable.
Mathematically speaking, the elliptic subproblems
$\big{(}\eqref{CH:INT:1}_{1},\eqref{CH:INT:3}\big{)}$ and
$\eqref{CH:INT:2}_{1}$ are coupled only through the trace relation (1.11c).

Compared to (1.12), we obtain the very different mass conservation law

$\displaystyle\int_{\Omega}u(t)\,\mathrm{d}x=\int_{\Omega}u(0)\,\mathrm{d}x\quad\text{and}\quad\int_{\Gamma}v(t)\,\mathrm{d}\Gamma=\int_{\Gamma}v(0)\,\mathrm{d}\Gamma,\quad
t\in[0,\infty),$ (1.14)

meaning that the bulk mass and the surface mass are conserved separately.
However, the energy dissipation law (1.13) is still satisfied by solutions of
this system.

The well-posedness of the LW model was addressed in [46, 33], and its long-
time behavior was investigated in [46, 50]. For numerical analysis and some
simulations we refer to [47, 4].

Moreover, a variant of the LW model (1.11) was proposed and investigated in
[41] where the relation between $u$ and $v$ is given by the Robin type
transmission condition

$\displaystyle K\partial_{\mathbf{n}}u=H(v)-u\quad\text{ on }\Sigma$

with $K>0$ and a function $H\in C^{2}(\mathbb{R})$ satisfying suitable growth
conditions. In particular, it was rigorously established in [41] that in the
case $H(s)=s$, solutions of this model converge to solutions of the LW model
in the limit $K\to 0$ in some suitable sense.

##### The case $0<L<\infty$ (KLLM model).

For $0<L<\infty$, the system (1.11) was recently proposed and analyzed by the
second author in collaboration with K.F. Lam, C. Liu and S. Metzger [42]. It
will thus be referred to as the KLLM model.

The Robin type condition (1.11c) establishes a connection between the GMS
model ((1.11) with $L=0$) and the LW model ((1.11) with $L=\infty$) despite
their very different chemical and physical properties. Suppose that $\beta>0$
and that $(u^{L},v^{L},\mu^{L},\theta^{L})$ is a solution of the system (1.11)
corresponding to the parameter $L>0$. Let $(u^{0},v^{0},\mu^{0},\theta^{0})$
denote its formal limit as $L\to 0$ and let
$(u^{\infty},v^{\infty},\mu^{\infty},\theta^{\infty})$ denote its formal limit
as $L\to\infty$. Passing to the limit in the Robin boundary condition, we
deduce that

$\displaystyle\beta\theta^{0}=\mu^{0}\quad\text{on
}\Sigma\quad\text{and}\quad\partial_{\mathbf{n}}\mu^{\infty}=0\quad\text{on
}\Sigma.$

This corresponds to the limit cases of instantaneous relaxation to chemical
equilibrium ($1/L\to\infty$), and the absence of adsorption and desorption
($1/L\to 0$). We infer that $(u^{0},v^{0},\mu^{0},\theta^{0})$ is a solution
to the GMS model while $(u^{\infty},v^{\infty},\mu^{\infty},\theta^{\infty})$
is a solution to the LW model. These formal considerations are rigorously
verified in [42]. In this regard, the Cahn–Hilliard system (1.11) with
$L\in(0,\infty)$ can be interpreted as an interpolation between the GMS model
and the LW model where the interpolation parameter $L$ corresponds to positive
but finite kinetic rates.

We observe that solutions of the KLLM model satisfy the same mass conservation
law (1.12) as solutions of the GMS model. However, we obtain an additional
term in the dissipation rate depending on the relaxation parameter $L$. To be
precise, it holds that

$\displaystyle\frac{d}{dt}E\big{(}u(t)\big{)}+m_{\Omega}\int_{\Omega}|\nabla\mu(t)|^{2}\,\mathrm{d}x+m_{\Gamma}\int_{\Gamma}|\nabla_{\Gamma}\theta(t)|^{2}\,\mathrm{d}\Gamma+\frac{m_{\Omega}}{L}\int_{\Gamma}\big{(}\beta\theta(t)-\mu(t)\big{)}^{2}\,\mathrm{d}\Gamma=0$
(1.15)

for all $t\in[0,\infty)$. In particular, this implies that the total free
energy $E$ is decreasing along solutions, and as $E$ is bounded from below (at
least for reasonable choices of $F$ and $G$), we infer that
$\frac{d}{dt}E(u(t))$ tends to zero as $t\to\infty$. As a consequence, the
chemical potentials will converge to the chemical equilibrium
$\mu=\beta\theta$ as $t\to\infty$.

We further point out that weak and strong well-posedness of the KLLM model was
established in [42]. As already mentioned above, the asymptotic limits $L\to
0$ and $L\to\infty$ were also discussed in [42]. We will exploit some of these
results for the limit $L\to 0$ in Section 5, Lemma 5.2.

For simulations and numerical analysis concerning the KLLM model, we refer to
[42, 5]. A nonlocal variant of the KLLM model (including a nonlocal dynamic
boundary condition) was proposed and investigated in [44].

##### Contents of this paper.

This paper is structured as follows. In Section 2, we first introduce some
notation and assumptions which will be used throughout the paper. We further
define some suitable spaces and operators and we introduce several important
interpolation inequalities. In Section 3, we present the well-posedness
results for both the KLLM model and the GMS model for the reader’s
convenience. Furthermore, we establish some properties of the weak solutions
to both models which are essential to investigate their long-time behavior.
Subsection 4.1 is devoted to the long-time analysis of the KLLM model. We
first characterize the set of stationary points containing the time-
independent solutions to the KLLM model. Next, we establish the existence of a
(unique) global attractor and we prove convergence of weak solutions to a
single stationary point as $t\to\infty$ by means of a Łojasiewicz–Simon
inequality. Eventually, we show that the global attractor can be represented
as the union of the unstable manifolds of all stationary points. In Subsection
4.2, supported by the results on long-time behavior that have already been
established in [36], we sketch how all results we proved in Subsection 4.1 for
the KLLM model can be obtained also for the GMS model in a similar fashion.
Next, in Section 5, we show that the global attractor of the GMS model (i.e.,
$L=0$) is stable with respect to perturbations of the kinetic rate (i.e.,
$L>0$ being small). Ultimately, in Section 6, we show that there exists a
family $\\{\mathfrak{M}^{L}\\}_{L\geq 0}$ of exponential attractors for the
system (1.11) such that the attractor $\mathfrak{M}^{0}$ associated with the
GMS model is robust in some certain sense against perturbations of the kinetic
rate (i.e., $L>0$ being small).

##### Comparison of the GMS/KLLM dynamics with the LW dynamics.

We point out that we do not see any possibility of transferring the results of
Section 5 and Section 6 to the scenario $L\to\infty$. This is mainly due to
the different mass conservation law of the LW model. Roughly speaking, the
mass conservation law (1.12) of the GMS/KLLM model fixes only one degree of
freedom, whereas the mass conservation law (1.14) of the LW model already
fixes two degrees of freedom. As a consequence, the GMS/KLLM model and the LW
model exhibit a different behavior regarding energy dissipation. Namely, as
only one degree of freedom is fixed by the mass conservation law, the free
energy can usually be decreased much further by the GMS/KLLM model than by the
LW model. This phenomenon can also be observed in numerical simulations, see,
e.g., [42, Fig. 4] or [5, Fig. 16]. This already indicates that the GMS/KLLM
model and the LW model differ in their long-time behavior. For more details
why our analysis in Section 5 and Section 6 fails in the situation
$L\to\infty$, we refer to Remark 5.6.

##### Further related results in the literature.

For further results on long-time behavior for Cahn–Hilliard models with
dynamic boundary conditions, we refer to [54, 35, 29, 50, 36, 13]. We also
want to mention [39] where the long-time dynamics of an Allen–Cahn model with
dynamic boundary conditions were studied, and [39, 13] where the long-time
behavior of a Caginalp phase field model with dynamic boundary conditions was
analyzed.

##  2 Preliminaries and important tools

We will now fix some notation and assumptions that are supposed to hold
throughout this paper.

##### Notation.

  1. $(\mathrm{N1})$

Sometimes, we will use the notation $\mathbb{R}_{0}^{+}:=[0,\infty)$. The
symbol $\mathbb{N}_{0}$ denotes the set of natural numbers including zero and
$\mathbb{N}=\mathbb{N}_{0}\setminus\\{0\\}$.

  2. $(\mathrm{N2})$

For any real numbers $k\geq 0$ and $1\leq p\leq\infty$, the standard Lebesgue
and Sobolev spaces over a domain $\Omega$ are denoted as $L^{p}(\Omega)$ and
$W^{k,p}(\Omega)$. We write $\|\cdot\|_{L^{p}(\Omega)}$ and
$\|\cdot\|_{W^{k,p}(\Omega)}$ to denote the standard norms on these spaces. If
$p=2$, these spaces are Hilbert spaces and we use the notation
$H^{k}(\Omega)=W^{k,2}(\Omega)$. We point out that $H^{0}(\Omega)$ can be
identified with $L^{2}(\Omega)$. An analogous notation is used for Lebesgue
and Sobolev spaces on $\Gamma:=\partial\Omega$, provided that the boundary is
sufficiently regular.

  3. $(\mathrm{N3})$

For any Banach space $X$, we write $X^{\prime}$ to denote its dual space. The
associated duality pairing of elements $\phi\in X^{\prime}$ and $\zeta\in X$
is denoted as $\langle\phi,\zeta\rangle_{X}$. If $X$ is a Hilbert space, we
write $(\cdot,\cdot)_{X}$ to denote its inner product.

  4. $(\mathrm{N4})$

We define

$\displaystyle\langle
u\rangle_{\Omega}:=\begin{cases}\frac{1}{\left|\Omega\right|}\langle
u,1\rangle_{H^{1}(\Omega)}&\text{ if }u\in H^{1}(\Omega)^{\prime},\\\
\frac{1}{\left|\Omega\right|}\int_{\Omega}u\,\mathrm{d}x&\text{ if }u\in
L^{1}(\Omega)\end{cases}$

to denote the (generalized) spatial mean of $u$. Here, $\left|\Omega\right|$
denotes the $d$-dimensional Lebesgue measure of $\Omega$. The spatial mean of
a function $v\in H^{1}(\Gamma)^{\prime}$ (or $v\in L^{1}(\Gamma)$,
respectively) is defined analogously.

  5. $(\mathrm{N5})$

Let $(X,d)$ be a metric space. Then for any sets $A,B\subset X$, the
_Hausdorff semidistance_ is defined as

$\displaystyle\mathrm{dist}_{X}(A,B):=\underset{a\in A}{\sup}\;\underset{b\in
B}{\inf}\;d\left(a,b\right),$ (2.1)

and the _symmetric Hausdorff distance_ is given by

$\displaystyle\mathrm{dist}_{\mathrm{sym},X}(A,B):=\max\big{(}\mathrm{dist}_{X}(A,B)\,{,}\,\mathrm{dist}_{X}(B,A)\big{)}.$
(2.2)

For any compact set $K\subset X$, the _fractal dimension of $K$_ is defined as

$\displaystyle\dim_{\mathrm{frac},X}(K):=\underset{r\to
0}{\lim\sup}\frac{\ln\big{(}\mathcal{N}_{r}(K;X)\big{)}}{-\ln(r)},$ (2.3)

where $\mathcal{N}_{r}(K;X)$ denotes the minimal number of balls in $X$ with
radius $r$ that are necessary to cover the set $K$.

##### Assumptions.

We make the following general assumptions.

  1. $(\mathrm{A1})$

We assume that $\Omega\subset\mathbb{R}^{d}$ with $d\in\\{2,3\\}$ is a bounded
domain whose boundary $\Gamma:=\partial\Omega$ is of class $C^{3}$. We further
use the notation

$\displaystyle
Q:=\Omega\times(0,\infty),\quad{\Sigma}:=\Gamma\times(0,\infty).$

  2. $(\mathrm{A2})$

In general, we assume that the constants occurring in the system (1.11)
satisfy $T$, $\beta$, $\kappa$, $m_{\Omega}$, $m_{\Gamma}>0$ and
$L\in[0,\infty)$. Since the choice of $\delta$, $\varepsilon$, $\kappa$,
$m_{\Omega}$ and $m_{\Gamma}$ has no impact on the mathematical analysis, we
will simply set $\delta=\varepsilon=\kappa=m_{\Omega}=m_{\Gamma}=1$ for
convenience in the mathematical analysis.

  3. $(\mathrm{A3})$

We assume that the potentials $F$ and $G$ are non-negative functions which can
be written as $F=F_{1}+F_{2}$ and $G=G_{1}+G_{2}$ with
$F_{1},F_{2},G_{1},G_{2}\in C^{2}(\mathbb{R})$ such that the following
properties hold:

    1. $(\mathrm{A3.1})$

There exist exponents $2<p\leq 4$ and $q>2$, as well as constants
$a_{F},a_{F^{\prime}},c_{F},c_{F^{\prime}}>0$ and $b_{F},b_{F^{\prime}}\geq 0$
such that for all $s\in\mathbb{R}$,

$\displaystyle a_{F}\left|s\right|^{p}-b_{F}$ $\displaystyle\leq F(s)$
$\displaystyle\leq c_{F}(1+\left|s\right|^{p}),$ (2.4a) $\displaystyle
a_{G}\left|s\right|^{q}-b_{G}$ $\displaystyle\leq G(s)$ $\displaystyle\leq
c_{G}(1+\left|s\right|^{q}),$ (2.4b) $\displaystyle
a_{F^{\prime}}\left|s\right|^{p-1}-b_{F^{\prime}}$
$\displaystyle\leq\left|F^{\prime}(s)\right|$ $\displaystyle\leq
c_{F^{\prime}}(1+\left|s\right|^{p-1}),$ (2.4c) $\displaystyle
a_{G^{\prime}}\left|s\right|^{q-1}-b_{G^{\prime}}$
$\displaystyle\leq\left|G^{\prime}(s)\right|$ $\displaystyle\leq
c_{G^{\prime}}(1+\left|s\right|^{q-1}).$ (2.4d)

    2. $(\mathrm{A3.2})$

The functions $F_{1}$ and $G_{1}$ are convex and non-negative. There further
exist positive constants $c_{F^{\prime\prime}}$ and $c_{G^{\prime\prime}}$
such that the second order derivatives satisfy the growth conditions

$\displaystyle 0\leq F_{1}^{\prime\prime}(s)\leq
c_{F^{\prime\prime}}(1+|s|^{p-2}),\quad 0\leq G_{1}^{\prime\prime}(s)\leq
c_{G^{\prime\prime}}(1+|s|^{q-2})$ (2.5)

for all $s\in\mathbb{R}$.

    3. $(\mathrm{A3.3})$

The derivatives $F_{2}^{\prime}$ and $G_{2}^{\prime}$ are Lipschitz
continuous. Consequently, there exist positive constants $d_{F}$, $d_{G}$,
$d_{F^{\prime}}$ and $d_{G^{\prime}}$ such that for all $s\in\mathbb{R}$,

$\displaystyle\left|F_{2}^{\prime}(s)\right|\leq
d_{F^{\prime}}(1+\left|s\right|),\quad$
$\displaystyle\left|G_{2}^{\prime}(s)\right|\leq
d_{G^{\prime}}(1+\left|s\right|),$ (2.6)
$\displaystyle\left|F_{2}(s)\right|\leq d_{F}(1+\left|s\right|^{2}),\quad$
$\displaystyle\left|G_{2}(s)\right|\leq d_{G}(1+\left|s\right|^{2}).$ (2.7)

In the analysis of the long-time dynamics, we will frequently have to use the
following additional assumption:

  1. $(\mathrm{A4})$

We assume that $F,G\in C^{\infty}(\mathbb{R})$ are analytic functions.

######  Remark 2.1.

We point out that the polynomial double-well potential

$\displaystyle W_{\text{dw}}(s)=\tfrac{1}{4}(s^{2}-1)^{2},\quad
s\in\mathbb{R},$

is a suitable choice for $F$ and $G$ as it satisfies ($\mathrm{A3}$) with
$p=4$ and $q=4$, and obviously also ($\mathrm{A4}$). However, singular
potentials like the logarithmic potential or the obstacle potential are not
admissible as they do not even satisfy ($\mathrm{A3}$).

##### Preliminaries.

We next introduce several function spaces, products, norms and operators that
will be used throughout this paper.

  1. $(\mathrm{P1})$

For any $k\in\mathbb{N}_{0}$ and any real number $p\in[1,\infty]$, we set

$\displaystyle\mathcal{L}^{p}:=L^{p}(\Omega)\times
L^{p}(\Gamma),\quad\text{and}\quad\mathcal{H}^{k}:=H^{k}(\Omega)\times
H^{k}(\Gamma),$

and we identify $\mathcal{L}^{2}$ with $\mathcal{H}^{0}$. Note that
$\mathcal{H}^{k}$ is a Hilbert space with respect to the inner product

$\displaystyle\big{(}(\phi,\psi),(\zeta,\xi)\big{)}_{\mathcal{H}^{k}}:=\big{(}\phi,\zeta\big{)}_{H^{k}(\Omega)}+\big{(}\psi,\xi\big{)}_{H^{k}(\Gamma)}\quad\text{for
all $(\phi,\psi),(\zeta,\xi)\in\mathcal{H}^{k}$,}$

and its induced norm
$\|\cdot\|_{\mathcal{H}^{k}}:=\left(\cdot,\cdot\right)_{\mathcal{H}^{k}}^{1/2}$.

  2. $(\mathrm{P2})$

For any $k\in\mathbb{N}$, we introduce the Hilbert space

$\displaystyle{\mathcal{V}^{k}}:=\big{\\{}(\phi,\psi)\in\mathcal{H}^{k}\;|\;\phi|_{\Gamma}=\psi\;\text{a.e.~{}on}\;\Gamma\big{\\}}$

endowed with the inner product
$\left(\cdot,\cdot\right)_{{\mathcal{V}^{k}}}:=\left(\cdot,\cdot\right)_{\mathcal{H}^{k}}$
and the norm $\|\cdot\|_{{\mathcal{V}^{k}}}:=\|\cdot\|_{\mathcal{H}^{k}}$.

  3. $(\mathrm{P3})$

For any $m\in\mathbb{R}$, $\beta>0$ and $k\in\mathbb{N}$, we set

$\displaystyle{\mathcal{W}_{\beta,m}^{k}}$
$\displaystyle:=\big{\\{}(\phi,\psi)\in{\mathcal{V}^{k}}\;|\;\beta\left|\Omega\right|\langle\phi\rangle_{\Omega}+\left|\Gamma\right|\langle\psi\rangle_{\Gamma}=m\big{\\}}.$

Endowed with the inner product

$\displaystyle\left(\cdot,\cdot\right)_{\mathcal{W}_{\beta,0}^{k}}:=\left(\cdot,\cdot\right)_{\mathcal{H}^{k}}$

and the induced norm, the space $\mathcal{W}_{\beta,0}^{k}$ is a Hilbert
space.

  4. $(\mathrm{P4})$

For $\beta>0$, we introduce the subspace

$\displaystyle{\mathcal{D}_{\beta}}:=\left\\{(\phi,\psi)\in\mathcal{H}^{1}\;\big{|}\;\phi|_{\Gamma}=\beta\psi\;\;\text{a.e.
on}\;\Gamma\right\\}\subset\mathcal{H}^{1}.$

Endowed with the inner product

$\displaystyle\left(\cdot,\cdot\right)_{\mathcal{D}_{\beta}}:=\left(\cdot,\cdot\right)_{\mathcal{H}^{1}}$

and its induced norm, the space ${\mathcal{D}_{\beta}}$ is a Hilbert space.
Moreover, we define the product

$\displaystyle\big{<}(\phi,\psi),(\zeta,\xi)\big{>}_{{\mathcal{D}_{\beta}}}:=\left(\phi,\zeta\right)_{L^{2}(\Omega)}+\left(\psi,\xi\right)_{L^{2}(\Gamma)}$

for all $(\phi,\psi),(\zeta,\xi)\in\mathcal{L}^{2}$. By means of the Riesz
representation theorem, this product can be extended to a duality pairing on
$({\mathcal{D}_{\beta}})^{\prime}\times{\mathcal{D}_{\beta}}$, which will also
be denoted as $\langle\cdot,\cdot\rangle_{{\mathcal{D}_{\beta}}}$.

In particular, the spaces
$\big{(}{\mathcal{D}_{\beta}},\mathcal{L}^{2},({\mathcal{D}_{\beta}})^{\prime}\big{)}$
form a Gelfand triplet, and the operator norm on
$({\mathcal{D}_{\beta}})^{\prime}$ is given by

$\displaystyle\|(\phi,\psi)\|_{({\mathcal{D}_{\beta}})^{\prime}}:=\sup\Big{\\{}\,\big{|}\langle(\phi,\psi),(\zeta,\xi)\rangle_{{\mathcal{D}_{\beta}}}\big{|}\;\Big{|}\;(\zeta,\xi)\in{\mathcal{D}_{\beta}}\text{
with }\|(\zeta,\xi)\|_{{\mathcal{D}_{\beta}}}=1\Big{\\}},$

for all $(\phi,\psi)\in({\mathcal{D}_{\beta}})^{\prime}$. Note that the
mapping

$\displaystyle\mathfrak{I}:{\mathcal{V}}^{1}\to{\mathcal{D}_{\beta}},\quad(\phi,\psi)\mapsto(\beta\phi,\psi)$

is an isomorphism.

  5. $(\mathrm{P5})$

Let $L\geq 0$, $\beta>0$ and $m\in\mathbb{R}$ be arbitrary. We define the
space

$\displaystyle\mathcal{H}^{1}_{L,\beta}$
$\displaystyle:=\begin{cases}\left\\{(\phi,\psi)\in\mathcal{H}^{1}\;|\;\beta\left|\Omega\right|\langle\phi\rangle_{\Omega}+\left|\Gamma\right|\langle\psi\rangle_{\Gamma}=0\right\\},&\text{if}\;L>0,\\\\[8.61108pt]
\left\\{(\phi,\psi)\in{\mathcal{D}_{\beta}}\;|\;\beta\left|\Omega\right|\langle\phi\rangle_{\Omega}+\left|\Gamma\right|\langle\psi\rangle_{\Gamma}=0\right\\},&\text{if}\;L=0,\end{cases}$

along with the inner product

$\displaystyle\big{(}(\phi,\psi),(\zeta,\xi)\big{)}_{L,\beta}$
$\displaystyle:=\int_{\Omega}\nabla\phi\cdot\nabla\zeta\,\mathrm{d}x+\int_{\Gamma}\nabla_{\Gamma}\psi\cdot\nabla_{\Gamma}\xi\,\mathrm{d}\Gamma$
$\displaystyle\qquad+\sigma(L)\int_{\Gamma}(\beta\psi-\phi)(\beta\xi-\zeta)\,\mathrm{d}\Gamma,$

for all $(\phi,\psi),(\zeta,\xi)\in\mathcal{H}^{1}_{L,\beta}$, where

$\displaystyle\sigma(L)=\begin{cases}L^{-1}&\text{if}\;L>0,\\\
0&\text{if}\;L=0.\end{cases}$ (2.8)

This is indeed an inner product since
$\left((\phi,\psi),(\phi,\psi)\right)_{L,\beta}=0$ already entails that $\phi$
and $\psi$ are constant almost everywhere and satisfy
$\phi|_{\Gamma}=\beta\psi$ almost everywhere on $\Gamma$. Then, the mean value
constraint of $\mathcal{H}^{1}_{L,\beta}$ directly yields $(\phi,\psi)=(0,0)$
almost everywhere. The induced norm is given as
$\|\,\cdot\,\|_{L,\beta}:=\left(\cdot,\cdot\right)_{L,\beta}^{1/2}$.

  6. $(\mathrm{P6})$

Let $L,\beta>0$ be arbitrary. We define the space

$\displaystyle\mathcal{H}_{\beta}^{-1}$
$\displaystyle:=\big{\\{}(\phi,\psi)\in(\mathcal{H}^{1})^{\prime}\;|\;\beta\left|\Omega\right|\langle\phi\rangle_{\Omega}+\left|\Gamma\right|\langle\psi\rangle_{\Gamma}=0\big{\\}}\subset(\mathcal{H}^{1})^{\prime}.$

We conclude from [43, Thm. 3.3] that for any
$(\phi,\psi)\in\mathcal{H}_{\beta}^{-1}$, there exists a unique weak solution
$\SS^{L}(\phi,\psi)=(\SS^{L}_{\Omega}(\phi,\psi),\SS^{L}_{\Gamma}(\phi,\psi))\in\mathcal{H}^{1}_{L,\beta}$
to the elliptic problem

$\displaystyle-\Delta\SS^{L}_{\Omega}$ $\displaystyle=-\phi$
$\displaystyle\text{ in }\Omega,$ (2.9a)
$\displaystyle-\Delta_{\Gamma}\SS^{L}_{\Gamma}+\beta\partial_{\bm{n}}\SS^{L}_{\Omega}$
$\displaystyle=-\psi$ $\displaystyle\text{ on }\Gamma,$ (2.9b) $\displaystyle
L\partial_{\bm{n}}\SS^{L}_{\Omega}$
$\displaystyle=(\beta\SS^{L}_{\Gamma}-\SS^{L}_{\Omega})$ $\displaystyle\text{
on }\Gamma.$ (2.9c)

This means that $\SS^{L}(\phi,\psi)$ satisfies the weak formulation

$\displaystyle\big{(}\SS^{L}(\phi,\psi),(\zeta,\xi)\big{)}_{L,\beta}=-\big{<}(\phi,\psi),(\zeta,\xi)\big{>}_{\mathcal{H}^{1}}$
(2.10)

for all test functions $(\zeta,\xi)\in\mathcal{H}^{1}$. As in [43, Thm. 3.3
and Cor. 3.5], we can thus define the solution operator

$\displaystyle\SS^{L}:\mathcal{H}_{\beta}^{-1}\to\mathcal{H}^{1}_{L,\beta},\quad(\phi,\psi)\mapsto\SS^{L}(\phi,\psi)=(\SS^{L}_{\Omega}(\phi),\SS^{L}_{\Gamma}(\phi,\psi))$

as well as an inner product and its induced norm on the space
$\mathcal{H}_{\beta}^{-1}$ by

$\displaystyle\big{(}(\phi,\psi),(\zeta,\xi)\big{)}_{L,\beta,*}$
$\displaystyle:=\big{(}\SS^{L}(\phi,\psi),\SS^{L}(\zeta,\xi)\big{)}_{L,\beta},$
$\displaystyle\|(\phi,\psi)\|_{L,\beta,*}$
$\displaystyle:=\left((\phi,\psi),(\phi,\psi)\right)_{L,\beta,*}^{1/2}$

for all $(\phi,\psi),(\zeta,\xi)\in\mathcal{H}_{\beta}^{-1}$. This norm is
equivalent to the standard operator norm on $\mathcal{H}_{\beta}^{-1}$. Since
$\mathcal{H}_{L,\beta}^{1}\subset\mathcal{H}_{\beta}^{-1}$, the product
$\left(\cdot,\cdot\right)_{L,\beta,*}$ can also be used as an inner product on
$\mathcal{H}_{L,\beta}^{1}$. Moreover, $\|\cdot\|_{L,\beta,*}$ is also a norm
on $\mathcal{H}_{L,\beta}^{1}$ but $\mathcal{H}_{L,\beta}^{1}$ is not complete
with respect to this norm.

  7. $(\mathrm{P7})$

Let $\beta>0$ be arbitrary. We define the space

$\displaystyle\mathcal{D}_{\beta}^{-1}$
$\displaystyle:=\Big{\\{}(\phi,\psi)\in({\mathcal{D}_{\beta}})^{\prime}\;\big{|}\;\big{<}(\phi,\psi),(\beta,1)\big{>}_{\mathcal{D}_{\beta}}=0\,\Big{\\}}\subset({\mathcal{D}_{\beta}})^{\prime}.$

Proceeding exactly as in [43, Proof of Thm. 3.3], we use the Lax–Milgram
theorem to show that for any $(\phi,\psi)\in\mathcal{D}_{\beta}^{-1}$, there
exists a unique weak solution
$\SS^{0}(\phi,\psi)=(\SS^{0}_{\Omega}(\phi),\SS^{0}_{\Gamma}(\phi,\psi))\in\mathcal{H}^{1}_{0,\beta}\subset{\mathcal{D}_{\beta}}$
to the elliptic problem

$\displaystyle-\Delta\SS^{0}_{\Omega}$ $\displaystyle=-\phi$
$\displaystyle\text{ in }\Omega,$ (2.11a)
$\displaystyle-\Delta_{\Gamma}\SS^{0}_{\Gamma}+\beta\partial_{\bm{n}}\SS^{0}_{\Omega}$
$\displaystyle=-\psi$ $\displaystyle\text{ on }\Gamma,$ (2.11b)
$\displaystyle\SS^{0}_{\Omega}|_{\Gamma}$
$\displaystyle=\beta\SS^{0}_{\Gamma}$ $\displaystyle\text{ on }\Gamma.$
(2.11c)

This means that $\SS^{0}(\phi,\psi)$ satisfies the weak formulation

$\displaystyle\big{<}\SS^{0}(\phi,\psi),(\zeta,\xi)\big{>}_{0,\beta}=-\big{<}(\phi,\psi),(\zeta,\xi)\big{>}_{{\mathcal{D}_{\beta}}}$
(2.12)

for all test functions $(\zeta,\xi)\in{\mathcal{D}_{\beta}}$. Similar to [43,
Thm. 3.3 and Cor. 3.5], we can thus define the solution operator

$\displaystyle\SS^{0}:\mathcal{D}_{\beta}^{-1}\to\mathcal{H}^{1}_{0,\beta},\quad(\phi,\psi)\mapsto\SS^{0}(\phi,\psi)=(\SS^{0}_{\Omega}(\phi),\SS^{0}_{\Gamma}(\phi,\psi))$

as well as an inner product and its induced norm on the space
$\mathcal{D}_{\beta}^{-1}$ by

$\displaystyle\big{(}(\phi,\psi),(\zeta,\xi)\big{)}_{0,\beta,*}$
$\displaystyle:=\big{(}\SS^{0}(\phi,\psi),\SS^{0}(\zeta,\xi)\big{)}_{0,\beta},$
$\displaystyle\|(\phi,\psi)\|_{0,\beta,*}$
$\displaystyle:=\left((\phi,\psi),(\phi,\psi)\right)_{0,\beta,*}^{1/2}$

for all $(\phi,\psi),(\zeta,\xi)\in\mathcal{D}_{\beta}^{-1}$. Since
$\mathcal{H}_{0,\beta}^{1}\subset\mathcal{D}_{\beta}^{-1}$, the product
$\left(\cdot,\cdot\right)_{L,\beta,*}$ can also be used as an inner product on
$\mathcal{H}_{0,\beta}^{1}$. Moreover, $\|\cdot\|_{L,\beta,*}$ is also a norm
on $\mathcal{H}_{0,\beta}^{1}$ but $\mathcal{H}_{0,\beta}^{1}$ is not complete
with respect to this norm.

We next show that for functions in ${\mathcal{W}_{\beta,0}^{1}}$, the norm
$\|\cdot\|_{L,\beta,*}$ can be bounded by the norm
$\|\cdot\|_{(\mathcal{H}^{1})^{\prime}}$ uniformly in $L\in[0,1]$.

######  Lemma 2.2.

Let $\beta>0$ be arbitrary. Then, there exists a constant $C>0$ depending only
on $\beta$ and $\Omega$ such that for all $L\in[0,1]$ and all
$(\phi,\psi)\in{\mathcal{W}_{\beta,0}^{1}}$, we have

$\displaystyle\|(\phi,\psi)\|_{L,\beta,*}\leq
C\|(\phi,\psi)\|_{(\mathcal{H}^{1})^{\prime}}.$

  * Proof.

Let $\beta>0$, $L\in[0,1]$ and $(\phi,\psi)\in{\mathcal{W}_{\beta,0}^{1}}$ be
arbitrary. In the following, the letter $C$ will denote generic positive
constants depending only on $\beta$ and $\Omega$. Recalling ($\mathrm{P6}$)
and ($\mathrm{P7}$), we obtain

$\displaystyle\|(\phi,\psi)\|_{L,\beta,*}^{2}$
$\displaystyle=\|\SS^{L}(\phi,\psi)\|_{L,\beta}^{2}=-\big{(}\SS^{L}(\phi,\psi),(\phi,\psi)\big{)}_{\mathcal{L}^{2}}$
$\displaystyle\leq\|\SS^{L}(\phi,\psi)\|_{\mathcal{H}^{1}}\,\|(\phi,\psi)\|_{(\mathcal{H}^{1})^{\prime}}.$
(2.13)

We can now apply [43, Cor. 7.2] (with $\alpha=\beta$ and $K=1$) to deduce that

$\displaystyle\|\SS^{L}(\phi,\psi)\|_{\mathcal{H}^{1}}\leq
C\,\|\SS^{L}(\phi,\psi)\|_{1,\beta}\;.$ (2.14)

Plugging this estimate into (Proof.), and recalling that $L\in[0,1]$, we
conclude that

$\displaystyle\|(\phi,\psi)\|_{L,\beta,*}^{2}$ $\displaystyle\leq
C\,\|\SS^{L}(\phi,\psi)\|_{1,\beta}\,\|(\phi,\psi)\|_{(\mathcal{H}^{1})^{\prime}}\leq
C\,\|\SS^{L}(\phi,\psi)\|_{L,\beta}\,\|(\phi,\psi)\|_{(\mathcal{H}^{1})^{\prime}}$
$\displaystyle=C\,\|(\phi,\psi)\|_{L,\beta,*}\,\|(\phi,\psi)\|_{(\mathcal{H}^{1})^{\prime}}.$
(2.15)

Since $(\phi,\psi)\in{\mathcal{W}_{\beta,0}^{1}}$ was arbitrary, the assertion
directly follows and the proof is complete. ∎

##### Interpolation inequalities.

We will further need the following interpolation estimates. It will turn out
to be essential that the constants in these estimates are independent of the
parameter $L$.

######  Lemma 2.3.

Suppose that ($\mathrm{A1}$) holds and let $L\geq 0$ and $\beta>0$ be
arbitrary. Then there exists a constant $c>0$ depending only on $\beta$, such
that for all $(u,v)\in{\mathcal{W}_{\beta,0}^{1}}\,$,

$\displaystyle\|(u,v)\|_{\mathcal{L}^{2}}^{2}\leq c\,\|(\nabla
u,\nabla_{\Gamma}v)\|_{\mathcal{L}^{2}}\left\|(u,v)\right\|_{L,\beta,*}.$
(2.16)

Moreover, for any $\alpha>0$ and all $(u,v)\in{\mathcal{W}_{\beta,0}^{1}}\,$,
it holds that

$\displaystyle\|(u,v)\|_{\mathcal{L}^{2}}^{2}\leq\alpha\|(\nabla
u,\nabla_{\Gamma}v)\|_{\mathcal{L}^{2}}^{2}+\frac{c^{2}}{4\alpha}\|(u,v)\|_{L,\beta,*}^{2}.$
(2.17)

  * Proof.

Let $L\in[0,\infty)$ and $(u,v)\in{\mathcal{W}_{\beta,0}^{1}}$ be arbitrary.
Recalling the definition of the operator $\SS^{L}$ (see ($\mathrm{P6}$) and
($\mathrm{P7}$)) and that $u|_{\Gamma}=v$ almost everywhere on $\Gamma$, we
obtain

$\displaystyle\min\\{\beta,1\\}\,\|(u,v)\|_{\mathcal{L}^{2}}\leq\beta\left\|u\right\|^{2}_{L^{2}(\Omega)}+\left\|v\right\|^{2}_{L^{2}(\Gamma)}$
$\displaystyle\quad=\left(u,\beta
u\right)_{L^{2}(\Omega)}+\left(v,v\right)_{L^{2}(\Omega)}$
$\displaystyle\quad=-\int_{\Omega}\nabla\big{(}\SS^{L}_{\Omega}(u,v)\big{)}\cdot\nabla(\beta
u)\,\mathrm{d}x-\int_{\Gamma}\nabla_{\Gamma}\big{(}\SS^{L}_{\Gamma}(u,v)\big{)}\cdot\nabla_{\Gamma}v\,\mathrm{d}\Gamma$
$\displaystyle\qquad\quad-\sigma(L)\int_{\Gamma}(\beta\SS^{L}_{\Gamma}(u,v)-\SS^{L}_{\Omega}(u,v))(\beta
v-\beta v)\,\mathrm{d}\Gamma$
$\displaystyle\quad\leq\beta\big{\|}\nabla\big{(}\SS^{L}_{\Omega}(u,v)\big{)}\big{\|}_{L^{2}(\Omega)}\big{\|}\nabla
u\big{\|}_{L^{2}(\Omega)}+\big{\|}\nabla_{\Gamma}\big{(}\SS^{L}_{\Gamma}(u,v)\big{)}\big{\|}_{L^{2}(\Gamma)}\big{\|}\nabla_{\Gamma}v\big{\|}_{L^{2}(\Gamma)}$
$\displaystyle\quad\leq\max\\{\beta,1\\}\,\|(\nabla
u,\nabla_{\Gamma}v)\|_{\mathcal{L}^{2}}\,\|\SS^{L}(u,v)\|_{L,\beta}$
$\displaystyle\quad=\max\\{\beta,1\\}\,\|(\nabla
u,\nabla_{\Gamma}v)\|_{\mathcal{L}^{2}}\,\|(u,v)\|_{L,\beta,*}\;.$

This proves (2.16) and thus, (2.17) direcly follows by means of Young’s
inequality. ∎

######  Lemma 2.4.

Suppose that ($\mathrm{A1}$) holds. Then there exist constants
$C_{1},C_{2},C_{3},C_{4}>0$ such that

$\displaystyle\|(u,v)\|_{\mathcal{H}^{1}}$ $\displaystyle\leq
C_{1}\|(u,v)\|_{\mathcal{H}^{2}}^{\frac{1}{2}}\|(u,v)\|_{\mathcal{L}^{2}}^{\frac{1}{2}}$
$\displaystyle\quad\text{for all $(u,v)\in\mathcal{H}^{2}$},$ (2.18)
$\displaystyle\|(u,v)\|_{\mathcal{H}^{2}}$ $\displaystyle\leq
C_{2}\|(u,v)\|_{\mathcal{H}^{3}}^{\frac{1}{2}}\|(u,v)\|_{\mathcal{H}^{1}}^{\frac{1}{2}}$
$\displaystyle\quad\text{for all $(u,v)\in\mathcal{H}^{3}$},$ (2.19)
$\displaystyle\|(u,v)\|_{\mathcal{H}^{1}}$ $\displaystyle\leq
C_{3}\|(u,v)\|_{\mathcal{H}^{3}}^{\frac{1}{3}}\|(u,v)\|_{\mathcal{L}^{2}}^{\frac{2}{3}}$
$\displaystyle\quad\text{for all $(u,v)\in\mathcal{H}^{3}$},$ (2.20)
$\displaystyle\|(u,v)\|_{\mathcal{H}^{2}}$ $\displaystyle\leq
C_{4}\|(u,v)\|_{\mathcal{H}^{3}}^{\frac{2}{3}}\|(u,v)\|_{\mathcal{L}^{2}}^{\frac{1}{3}}$
$\displaystyle\quad\text{for all $(u,v)\in\mathcal{H}^{3}$}.$ (2.21)

  * Proof.

To prove (2.18), let $(u,v)\in\mathcal{H}^{2}$ be arbitrary. It obviously
holds that

$\displaystyle\|(u,v)\|_{\mathcal{L}^{2}}^{2}\leq\|(u,v)\|_{\mathcal{H}^{2}}\|(u,v)\|_{\mathcal{L}^{2}}.$
(2.22)

Using the Gagliardo–Nierenberg inequality, we obtain

$\displaystyle\|\nabla
u\|_{L^{2}(\Omega)}^{2}\leq\|u\|_{H^{2}(\Omega)}\|u\|_{L^{2}(\Omega)}\leq\|(u,v)\|_{\mathcal{H}^{2}}\|(u,v)\|_{\mathcal{L}^{2}}.$
(2.23)

Moreover, the Gagliardo–Nierenberg inequality for compact Riemannian manifolds
(see, e.g., [3, Thm. 3.70]) implies that

$\displaystyle\|\nabla_{\Gamma}v\|_{L^{2}(\Gamma)}^{2}\leq\|v\|_{H^{2}(\Gamma)}\|v\|_{L^{2}(\Gamma)}\leq\|(u,v)\|_{\mathcal{H}^{2}}\|(u,v)\|_{\mathcal{L}^{2}}.$
(2.24)

Now, adding (2.22), (2.23) and (2.24) proves (2.18).

The inequality (2.19) follows by applying (2.18) on the components
$\big{(}[\nabla u]_{i},[\nabla_{\Gamma}v]_{i}\big{)}$, $i=1,2,3$. Eventually,
the estimates (2.20) and (2.21) are direct consequences of (2.18) and (2.19).
This completes the proof. ∎

The following result is a direct consequence of Lemma 2.3 and Lemma 2.4.

######  Corollary 2.5.

Suppose that ($\mathrm{A1}$) holds and let $L\geq 0$ and $\beta>0$ be
arbitrary. Then there exist positive constants $c_{1}$ and $c_{2}$ depending
only on $\beta$ such that

$\displaystyle\|(u,v)\|_{\mathcal{H}^{1}}$ $\displaystyle\leq
c_{1}\,\|(u,v)\|_{\mathcal{H}^{2}}^{\frac{2}{3}}\|(u,v)\|_{L,\beta,*}^{\frac{1}{3}}$
for all $(u,v)\in\mathcal{W}_{\beta,0}^{2}$, (2.25)
$\displaystyle\|(u,v)\|_{\mathcal{H}^{1}}$ $\displaystyle\leq
c_{2}\,\|(u,v)\|_{\mathcal{H}^{3}}^{\frac{1}{2}}\|(u,v)\|_{L,\beta,*}^{\frac{1}{2}}$
for all $(u,v)\in\mathcal{W}_{\beta,0}^{3}$. (2.26)

##  3 Well-posedness and further essential properties

In this section, we present the weak well-posedness results for the problem
(1.11) with $L\in[0,\infty)$, and we show that these weak solutions satisfy a
certain parabolic smoothing property. For simplicity, and as their choice has
no impact on the analysis we will carry out, the positive constants
$m_{\Omega}$, $m_{\Gamma}$, $\varepsilon$, $\delta$ and $\kappa$ are set to
one. By this choice, the system (1.11) can be restated as follows.

$\displaystyle\partial_{t}u=\Delta\mu,$ $\displaystyle\mu=-\Delta
u+F^{\prime}(u)$ $\displaystyle\quad\text{in }Q,$ (3.1a)
$\displaystyle\partial_{t}v=\Delta_{\Gamma}\theta-\beta\partial_{\bm{n}}\mu,$
$\displaystyle\theta=-\Delta_{\Gamma}v+G^{\prime}(v)+\partial_{\bm{n}}u$
$\displaystyle\quad\text{on }\Sigma,$ (3.1b) $\displaystyle u|_{\Gamma}=v$
$\displaystyle\quad\text{on }\Sigma,$ (3.1c)
$\displaystyle\left\\{\begin{aligned} \mu|_{\Sigma}&=\beta\theta\\\
L\partial_{\bm{n}}\mu|_{\Sigma}&=\beta\theta-\mu|_{\Sigma}\end{aligned}\right.$
$\displaystyle\quad\begin{aligned} &\text{if}\;L=0,\\\
&\text{if}\;L\in(0,\infty),\end{aligned}$ $\displaystyle\quad\text{on
}\Sigma,$ (3.1d) $\displaystyle(u,v)|_{t=0}=(u_{0},v_{0})$
$\displaystyle\text{with}\;u_{0}|_{\Sigma}=v_{0}$ $\displaystyle\quad\text{in
}\Omega\times\Gamma.$ (3.1e)

The total free energy associated with this system is given as

$\displaystyle E(u,v)=\int_{\Omega}\frac{1}{2}\left|\nabla
u\right|^{2}+F(u)\,\mathrm{d}x+\int_{\Gamma}\frac{1}{2}\left|\nabla_{\Gamma}v\right|^{2}+G(v)\,\mathrm{d}\Gamma.$
(3.2)

In the following, we will discuss the cases $0<L<\infty$ (KLLM model) and
$L=0$ (GMS model) separately.

###  3.1 The KLLM model ($0<L<\infty$)

######  Theorem 3.1 (Weak well-posedness and smoothing property for the KLLM
model).

Suppose that ($\mathrm{A1}$)–($\mathrm{A3}$) hold and that $L\in(0,\infty)$.
Let $m\in\mathbb{R}$ be arbitrary and let
$(u_{0},v_{0})\in{\mathcal{W}_{\beta,m}^{1}}$ be any initial datum.

Then there exists a unique global weak solution
$(u^{L},v^{L},\mu^{L},\theta^{L})$ of the system (3.1) existing on the whole
time interval $\mathbb{R}_{0}^{+}$ and having the following properties:

  1. $(\mathrm{i})$

For every $T>0$, the solution has the regularity

$\displaystyle\begin{aligned} \big{(}u^{L},v^{L}\big{)}&\in
C^{0,\frac{1}{4}}([0,T];\mathcal{L}^{2})\cap
L^{\infty}(0,T;{\mathcal{V}}^{1})\\\ &\qquad\cap
L^{2}(0,T;\mathcal{H}^{3})\cap
H^{1}\big{(}0,T;(\mathcal{H}^{1})^{\prime}\big{)},\end{aligned}$ (3.3a)
$\displaystyle\big{(}\mu^{L},\theta^{L}\big{)}\in L^{2}(0,T;\mathcal{H}^{1}).$
(3.3b)

  2. $(\mathrm{ii})$

The solution satisfies

$\displaystyle\big{<}\partial_{t}u^{L},w\big{>}_{H^{1}(\Omega)}$
$\displaystyle=-\int_{\Omega}\nabla\mu^{L}\cdot\nabla
w\,\mathrm{d}x+\int_{\Gamma}\tfrac{1}{L}(\beta\theta^{L}-\mu^{L})w\,\mathrm{d}\Gamma,$
(3.4a) $\displaystyle\big{<}\partial_{t}v^{L},z\big{>}_{H^{1}(\Gamma)}$
$\displaystyle=-\int_{\Gamma}\nabla_{\Gamma}\theta^{L}\cdot\nabla_{\Gamma}z\,\mathrm{d}\Gamma-\int_{\Gamma}\tfrac{1}{L}(\beta\theta^{L}-\mu^{L})\beta
z\,\mathrm{d}\Gamma,$ (3.4b) a.e. on $\mathbb{R}_{0}^{+}$ for all test
functions $(w,z)\in\mathcal{H}^{1}$, and $\displaystyle\mu^{L}=-\Delta
u^{L}+F^{\prime}(u^{L})$ $\displaystyle\quad\text{a.e.~{}in Q},$ (3.4c)
$\displaystyle\theta^{L}=-\Delta_{\Gamma}v^{L}+G^{\prime}(v^{L})+\partial_{\mathbf{n}}u^{L},$
$\displaystyle\quad\text{a.e.~{}in $\Sigma$},$ (3.4d) $\displaystyle
u^{L}|_{\Sigma}=v^{L},$ $\displaystyle\quad\text{a.e.~{}in $\Sigma$},$ (3.4e)
$\displaystyle(u^{L},v^{L})|_{t=0}=(u_{0},v_{0})$
$\displaystyle\quad\text{a.e.~{}in }\Omega\times\Gamma.$ (3.4f)

  3. $(\mathrm{iii})$

The solution satisfies the global mass conservation law

$\displaystyle\beta\int_{\Omega}u^{L}(t)\,\mathrm{d}x+\int_{\Gamma}v^{L}(t)\,\mathrm{d}\Gamma=\beta\int_{\Omega}u_{0}\,\mathrm{d}x+\int_{\Gamma}v_{0}\,\mathrm{d}\Gamma=m\quad$
(3.5)

for all $t\in\mathbb{R}_{0}^{+}$, i.e.,
$\big{(}u^{L}(t),v^{L}(t)\big{)}\in{\mathcal{W}_{\beta,m}^{1}}$ for almost all
$t\in\mathbb{R}_{0}^{+}$.

  4. $(\mathrm{iv})$

The solution satisfies the energy inequality

$\displaystyle
E\big{(}u^{L}(t),v^{L}(t)\big{)}+\frac{1}{2}\int_{0}^{t}\big{\|}\big{(}\mu^{L}(s),\theta^{L}(s)\big{)}\big{\|}_{L,\beta}^{2}\,\mathrm{d}s\leq
E(u_{0},v_{0})$ (3.6)

for almost all $t\geq 0$.

  5. $(\mathrm{v})$

The solution satisfies the smoothing property, i.e., there exists a constant
$C_{*}>0$ depending only on $\Omega$, $\beta$, $F$, $G$ and
$\|(u_{0},v_{0})\|_{\mathcal{H}^{1}}$, such that

$\displaystyle\big{\|}\big{(}u^{L}(t),v^{L}(t)\big{)}\big{\|}_{\mathcal{H}^{3}}\leq
C_{*}\left(\dfrac{1+t}{t}\right)^{\frac{1}{2}}$ (3.7)

for almost all $t\geq 0$.

######  Remark 3.2.

We point out that the above results of Theorem 3.1 hold true, if the bounds
from below in the conditions (2.4a) and (2.4b) are omitted. This means, it
would be sufficient to demand that

$\displaystyle F(s)\leq c_{F}(1+\left|s\right|^{p})\quad\text{and}\quad
G(s)\leq c_{G}(1+\left|s\right|^{q})$

for all $s\in\mathbb{R}$, even though the stronger assumptions were used in
[42] to which we will refer in the subsequent proof. This is because instead
of [42, Lem. 2.1], the stronger Poincaré type inequality

$\displaystyle\|(u,v)\|_{\mathcal{L}^{2}}\leq C_{P}\|\nabla
u\|_{L^{2}(\Omega)}\quad\text{for all $(u,v)\in{\mathcal{W}_{\beta,0}^{1}}$}$
(3.8)

could be used to prove the existence of a weak solution. The proof of (3.8)
can be done similarly to the proof of [42, Lem. 2.1] via a contradiction
argument. It relies on the fact that if $\nabla u=0$ a.e. in $\Omega$, then
the functions $u$ and $v=u|_{\Gamma}$ must be constant a.e. in $\Omega$ and
a.e. on $\Gamma$, respectively. Then, the mean value condition
$\beta\left|\Omega\right|\langle u\rangle_{\Omega}+\left|\Gamma\right|\langle
v\rangle_{\Gamma}=0$ ensures that $u=0$ a.e. in $\Omega$ and $v=0$ a.e. on
$\Gamma$.

Nevertheless, in Assumption $\mathrm{A3}$, we stick to the stronger conditions
as demanded in [42] since such kind of estimates are very useful in the proof
of Lemma 4.4 (see (4.13) and (4.14)).

We further point out that the above comments also apply for the GMS model
($L=0$) whose well-posedness will be discussed in Subsection 3.2.

  * Proof of Theorem 3.1.

The existence and uniqueness of a global unique weak solution which satisfies
the assertions (i)–(iv) has already been established in [42, Thm. 3.1].

Hence, only the smoothing property (v) remains to be proved. In the following,
we omit the superscript $L$ to provide a cleaner presentation. Let us fix
arbitrary test functions $(w,z)\in\mathcal{H}^{1}$ be arbitrary. Adding (3.4a)
and (3.4b), we obtain

$\displaystyle\big{\langle}\partial_{t}u,w\big{\rangle}_{H^{1}(\Omega)}+\big{\langle}\partial_{t}v,z\big{\rangle}_{H^{1}(\Gamma)}=-\big{(}(\mu,\theta),(w,z)\big{)}_{L,\beta}\quad\text{a.e.~{}on
$\mathbb{R}_{0}^{+}$}$ (3.9)

for all test functions $(w,z)\in\mathcal{H}^{1}$.

In the following, let $C$ denote a generic positive constant depending only on
$\Omega$, $\beta$, $F$, $G$ and $\|(u_{0},v_{0})\|_{\mathcal{H}^{1}}$ that may
change its value from line to line. For any $\tau\in\mathbb{R}_{0}^{+}$ and
$h>0$, we will write

$\displaystyle\partial_{t}^{h}f(\tau)=\frac{1}{h}\big{(}f(\tau+h)-f(\tau)\big{)}$
(3.10)

to denote the forward-in-time difference quotient of a suitable function $f$.

From (3.9), we deduce that

$\displaystyle\big{\langle}\partial_{t}^{h}\partial_{t}u(\tau),w\big{\rangle}_{H^{1}(\Omega)}+\big{\langle}\partial_{t}^{h}\partial_{t}v(\tau),z\big{\rangle}_{H^{1}(\Gamma)}=-\Big{(}\big{(}\partial_{t}^{h}\mu(\tau),\partial_{t}^{h}\theta(\tau)\big{)},(w,z)\Big{)}_{L,\beta}$
(3.11)

for almost all $\tau\in\mathbb{R}_{0}^{+}$ where

$\displaystyle\partial_{t}^{h}\mu(\tau)$
$\displaystyle=-\Delta\partial_{t}^{h}u(\tau)+\frac{1}{h}\Big{(}F^{\prime}\big{(}u(\tau+h)\big{)}-F^{\prime}\big{(}u(\tau)\big{)}\Big{)}$
a.e. in $\Omega$, (3.12) $\displaystyle\partial_{t}^{h}\theta(\tau)$
$\displaystyle=-\Delta_{\Gamma}\partial_{t}^{h}v(\tau)+\frac{1}{h}\Big{(}G^{\prime}\big{(}v(\tau+h)\big{)}-G^{\prime}\big{(}v(\tau)\big{)}\Big{)}+\partial_{n}\partial_{t}^{h}u(\tau)$
a.e. on $\Gamma$ (3.13)

due to (3.4c) and (3.4d). Plugging
$(w,z)=\SS^{L}\big{(}\partial_{t}^{h}u(\tau),\partial_{t}^{h}v(\tau)\big{)}$
into (3.11), recalling (3.12) and (3.13), and using integration by parts, we
obtain

$\displaystyle-\dfrac{1}{2}\dfrac{\mathrm{d}}{\mathrm{d}\tau}\big{\|}\big{(}\partial_{t}^{h}u(\tau),\partial_{t}^{h}v(\tau)\big{)}\big{\|}_{L,\beta,*}^{2}=-\Big{(}\big{(}\partial_{t}^{h}\mu(\tau),\partial_{t}^{h}\theta(\tau)\big{)},\SS^{L}\big{(}\partial_{t}^{h}u(\tau),\partial_{t}^{h}v(\tau)\big{)}\Big{)}_{L,\beta}$
$\displaystyle\quad=\big{(}\partial_{t}^{h}\mu(\tau),\partial_{t}^{h}u(\tau)\big{)}_{L^{2}(\Omega)}+\big{(}\partial_{t}^{h}\theta(\tau),\partial_{t}^{h}v(\tau)\big{)}_{L^{2}(\Gamma)}$
$\displaystyle\quad=\big{\|}\nabla\partial_{t}^{h}u(\tau)\big{\|}^{2}_{L^{2}(\Omega)}+\dfrac{1}{h}\int_{\Omega}\Big{(}F^{\prime}\big{(}u(\tau+h)\big{)}-F^{\prime}\big{(}u(\tau)\big{)}\Big{)}\partial_{t}^{h}u(\tau)\,\mathrm{d}x$
$\displaystyle\qquad+\big{\|}\nabla_{\Gamma}\partial_{t}^{h}v(\tau)\big{\|}^{2}_{L^{2}(\Gamma)}+\dfrac{1}{h}\int_{\Gamma}\Big{(}G^{\prime}\big{(}v(\tau+h)\big{)}-G^{\prime}\big{(}v(\tau)\big{)}\Big{)}\partial_{t}^{h}v(\tau)\,\mathrm{d}\Gamma$

for almost all $\tau\in\mathbb{R}_{0}^{+}$. Hence, we have

$\displaystyle\dfrac{1}{2}\dfrac{\mathrm{d}}{\mathrm{d}\tau}\big{\|}\big{(}\partial_{t}^{h}u(\tau),\partial_{t}^{h}v(\tau)\big{)}\big{\|}_{L,\beta,*}^{2}+\big{\|}\nabla\partial_{t}^{h}u(\tau)\big{\|}^{2}_{L^{2}(\Omega)}+\big{\|}\nabla_{\Gamma}\partial_{t}^{h}v(\tau)\big{\|}^{2}_{L^{2}(\Gamma)}$
$\displaystyle\qquad+\dfrac{1}{h}\int_{\Omega}\Big{(}F_{1}^{\prime}\big{(}u(\tau+h)\big{)}-F_{1}^{\prime}\big{(}u(\tau)\big{)}\Big{)}\partial_{t}^{h}u(\tau)\,\mathrm{d}x$
$\displaystyle\qquad+\dfrac{1}{h}\int_{\Gamma}\Big{(}G_{1}^{\prime}\big{(}v(\tau+h)\big{)}-G_{1}^{\prime}\big{(}v(\tau)\big{)}\Big{)}\partial_{t}^{h}v(\tau)\,\mathrm{d}\Gamma$
$\displaystyle=-\dfrac{1}{h}\int_{\Omega}\Big{(}F_{2}^{\prime}\big{(}u(\tau+h)\big{)}-F_{2}^{\prime}\big{(}u(\tau)\big{)}\Big{)}\partial_{t}^{h}u(\tau)\,\mathrm{d}x$
$\displaystyle\qquad-\dfrac{1}{h}\int_{\Gamma}\Big{(}G_{2}^{\prime}\big{(}v(\tau+h)\big{)}-G_{2}^{\prime}\big{(}v(\tau)\big{)}\Big{)}\partial_{t}^{h}v(\tau)\,\mathrm{d}\Gamma$

for all $\tau\in\mathbb{R}_{0}^{+}$. Recall that $F_{2}$, $G_{2}$ are
Lipschitz continuous, and that $F_{1}$ and $G_{1}$ are convex which directly
implies that $F_{1}^{\prime}$ and $G_{1}^{\prime}$ are monotone. Hence, we
obtain

$\displaystyle\begin{aligned}
&\dfrac{1}{2}\dfrac{\mathrm{d}}{\mathrm{d}\tau}\big{\|}\big{(}\partial_{t}^{h}u(\tau),\partial_{t}^{h}v(\tau)\big{)}\big{\|}_{L,\beta,*}^{2}+\big{\|}\nabla\partial_{t}^{h}u(\tau)\big{\|}^{2}_{L^{2}(\Omega)}+\big{\|}\nabla_{\Gamma}\partial_{t}^{h}v(\tau)\big{\|}^{2}_{L^{2}(\Gamma)}\\\
&\quad\leq
C\big{(}\big{\|}\partial_{t}^{h}u(\tau)\big{\|}^{2}_{L^{2}(\Omega)}+\big{\|}\partial_{t}^{h}v(\tau)\big{\|}^{2}_{L^{2}(\Gamma)}\big{)}\end{aligned}$
(3.14)

for all $\tau\in\mathbb{R}_{0}^{+}$. Using Lemma 2.3 we conclude that

$\displaystyle\begin{aligned}
&\dfrac{\mathrm{d}}{\mathrm{d}\tau}\big{\|}\big{(}\partial_{t}^{h}u(\tau),\partial_{t}^{h}v(\tau)\big{)}\big{\|}_{L,\beta,*}^{2}+\big{\|}\nabla\partial_{t}^{h}u(\tau)\big{\|}^{2}_{L^{2}(\Omega)}+\big{\|}\nabla_{\Gamma}\partial_{t}^{h}v(\tau)\big{\|}^{2}_{L^{2}(\Gamma)}\\\
&\quad\leq
C\big{\|}\big{(}\partial_{t}^{h}u(\tau),\partial_{t}^{h}v(\tau)\big{)}\big{\|}_{L,\beta,*}^{2}\end{aligned}$
(3.15)

for all $\tau\in\mathbb{R}_{0}^{+}$. Multiplying both sides by $\tau$ and
integrating the resulting inequality with respect to $\tau$ from $0$ to $t$,
we obtain

$\displaystyle
t\big{\|}\big{(}\partial_{t}^{h}u(t),\partial_{t}^{h}v(t)\big{)}\big{\|}_{L,\beta,*}^{2}\leq\int_{0}^{t}\big{(}C\tau+1\big{)}\big{\|}\big{(}\partial_{t}^{h}u(\tau),\partial_{t}^{h}v(\tau)\big{)}\big{\|}_{L,\beta,*}^{2}\,\mathrm{d}\tau$
(3.16)

for all $t\in\mathbb{R}_{0}^{+}$. Moreover, we have

$\displaystyle\big{\|}\big{(}\partial_{t}^{h}u(\tau),\partial_{t}^{h}v(\tau)\big{)}\big{\|}_{L,\beta,*}^{2}\leq\dfrac{1}{h}\int_{\tau}^{\tau+h}\big{\|}\big{(}\partial_{t}u(s),\partial_{t}v(s)\big{)}\big{\|}_{L,\beta,*}^{2}\,\mathrm{d}s,$
(3.17) $\displaystyle\lim_{h\to
0}\dfrac{1}{h}\int_{\tau}^{\tau+h}\big{\|}\big{(}\partial_{t}u(s),\partial_{t}v(s)\big{)}\big{\|}_{L,\beta,*}^{2}\,\mathrm{d}s=\big{\|}\big{(}\partial_{t}u(\tau),\partial_{t}v(\tau)\big{)}\big{\|}_{L,\beta,*}^{2}.$
(3.18)

Recalling the energy inequality (3.6) and the fact that the energy $E$ is non-
negative (since the potentials $F$ and $G$ are non-negative), we derive the
estimate

$\displaystyle\int_{0}^{t}\big{\|}\nabla\mu(\tau)\big{\|}_{L^{2}(\Omega)}^{2}+\big{\|}\nabla_{\Gamma}\theta(\tau)\big{\|}_{L^{2}(\Gamma)}^{2}+\frac{1}{L}\big{\|}\beta\theta(\tau)-\mu(\tau)\big{\|}^{2}_{L^{2}(\Gamma)}\,\mathrm{d}\tau\leq
C.$ (3.19)

By the definition of the solution operator $\SS^{L}$, we infer from (3.4a) and
(3.4b) that

$\displaystyle\mu(\tau)=\SS^{L}_{\Omega}\big{(}\partial_{t}u(\tau),\partial_{t}v(\tau)\big{)}+c(\tau)\beta,\quad\theta(\tau)=\SS^{L}_{\Gamma}\big{(}\partial_{t}u(\tau),\partial_{t}v(\tau)\big{)}+c(\tau)$
(3.20)

for almost all $\tau\in\mathbb{R}_{0}^{+}$ and some function $c=c(\tau)$.
Testing (3.4a) and (3.4b) with
$\SS^{L}\big{(}\partial_{t}u(\tau),\partial_{t}v(\tau)\big{)}$, and adding the
resulting equations, we infer that

$\displaystyle\begin{aligned}
&\big{\|}\big{(}\partial_{t}u(\tau),\partial_{t}v(\tau)\big{)}\big{\|}_{L,\beta,*}^{2}=\big{\|}\big{(}\mu(\tau),\theta(\tau)\big{)}\big{\|}_{L,\beta}^{2}\\\
&\quad=\big{\|}\nabla\mu(\tau)\big{\|}_{L^{2}(\Omega)}^{2}+\big{\|}\nabla_{\Gamma}\theta(\tau)\big{\|}_{L^{2}(\Gamma)}^{2}+\frac{1}{L}\big{\|}\beta\theta(\tau)-\mu(\tau)\big{\|}^{2}_{L^{2}(\Gamma)}\end{aligned}$
(3.21)

for almost all $\tau\in\mathbb{R}$. Consequently, after integrating with
respect to $\tau$ from $0$ to $t$, (3.19) and (3.21) imply

$\displaystyle\int_{0}^{t}\big{\|}\big{(}\partial_{t}u(\tau),\partial_{t}v(\tau)\big{)}\big{\|}_{L,\beta,*}^{2}\,\mathrm{d}\tau\leq
C\quad\text{for all $t\in\mathbb{R}_{0}^{+}$.}$ (3.22)

Therefore, recalling (3.17), (3.18) and (3.22), we obtain from (3.16) that

$\displaystyle
t\big{\|}\big{(}\partial_{t}^{h}u(t),\partial_{t}^{h}v(t)\big{)}\big{\|}_{L,\beta,*}^{2}\leq
C\big{(}t+1\big{)}\quad\text{for all $t\in\mathbb{R}_{0}^{+}$.}$

Invoking Lebesgue’s convergence theorem and passing to the limit $h\to 0$, we
get

$\displaystyle\big{\|}\big{(}\partial_{t}u(t),\partial_{t}v(t)\big{)}\big{\|}_{L,\beta,*}^{2}\leq
C\left(\dfrac{t+1}{t}\right)\quad\text{for all $t\in\mathbb{R}_{0}^{+}$},$
(3.23)

which in turn yields

$\displaystyle\big{\|}\nabla\mu(t)\big{\|}_{L^{2}(\Omega)}^{2}+\big{\|}\nabla_{\Gamma}\theta(t)\big{\|}_{L^{2}(\Gamma)}^{2}\leq
C\left(\dfrac{t+1}{t}\right)\quad\text{for almost all
$t\in\mathbb{R}_{0}^{+}$}$ (3.24)

because of (3.21). Proceeding as in [42, Proof of Thm. 3.1, Step 3] we obtain
the estimates

$\displaystyle\big{\|}\mu(t)\big{\|}_{L^{2}(\Omega)}$ $\displaystyle\leq
C\big{(}1+\big{\|}\nabla\mu(t)\big{\|}_{L^{2}(\Omega)}\big{)},$ (3.25)
$\displaystyle\big{\|}\theta(t)\big{\|}_{L^{2}(\Gamma)}$ $\displaystyle\leq
C\big{(}1+\big{\|}\nabla\mu(t)\big{\|}_{L^{2}(\Omega)}+\big{\|}\nabla_{\Gamma}\theta(t)\big{\|}_{L^{2}(\Gamma)}\big{)}$
(3.26)

for almost all $t\in\mathbb{R}_{0}^{+}$. This directly implies that

$\displaystyle\big{\|}\big{(}\mu(t),\theta(t)\big{)}\big{\|}_{\mathcal{H}^{1}}\leq
C\left(\dfrac{t+1}{t}\right)^{\frac{1}{2}}$ (3.27)

for almost all $t\in\mathbb{R}_{0}^{+}$. Furthermore, recalling the growth
assumptions on $F$ and $G$ imposed in ($\mathrm{A3}$), we conclude from the
energy inequality (3.6) that

$\displaystyle\big{\|}\big{(}u(t),v(t)\big{)}\big{\|}_{\mathcal{H}^{1}}\leq
C+CE\big{(}u(t),v(t)\big{)}\leq C+CE\big{(}u_{0},v_{0}\big{)}\leq C$ (3.28)

for almost all $t\in\mathbb{R}_{0}^{+}$. Since $(u,v)\in
L^{2}(0,T;\mathcal{H}^{3})$ for all $T>0$, we have

$\displaystyle-\Delta u(t)$
$\displaystyle=\mu(t)-F^{\prime}\big{(}u(t)\big{)}$
$\displaystyle\quad\text{a.e.~{}in $\Omega$},$ (3.29)
$\displaystyle-\Delta_{\Gamma}v(t)+\partial_{n}u(t)$
$\displaystyle=\theta(t)-G^{\prime}\big{(}v(t)\big{)}$ a.e. on $\Gamma$ (3.30)

for almost all $t\in\mathbb{R}_{0}^{+}$. Invoking the growth assumptions on
$F$ and $G$ in ($\mathrm{A3}$), recalling that $p\leq 4$, and using the
continuous embeddings $H^{1}(\Omega)\hookrightarrow L^{6}(\Omega)$ and
$H^{1}(\Gamma)\hookrightarrow L^{r}(\Gamma)$ for all $r\in[1,\infty)$, we
further deduce that for almost all $t\in\mathbb{R}_{0}^{+}$,

$\displaystyle\big{\|}F^{\prime}\big{(}u(t)\big{)}\big{\|}_{L^{2}(\Omega)}$
$\displaystyle\leq C+C\|u(t)\|_{H^{1}(\Omega)}^{p-1}\leq C,$
$\displaystyle\big{\|}G^{\prime}\big{(}v(t)\big{)}\big{\|}_{L^{2}(\Gamma)}$
$\displaystyle\leq C+C\|v(t)\|_{H^{1}(\Gamma)}^{q-1}\leq C,$
$\displaystyle\big{\|}F^{\prime\prime}\big{(}u(t)\big{)}\big{\|}_{L^{3}(\Omega)}$
$\displaystyle\leq C+C\|u(t)\|_{H^{1}(\Omega)}^{p-2}\leq C,$
$\displaystyle\big{\|}G^{\prime\prime}\big{(}v(t)\big{)}\big{\|}_{L^{3}(\Gamma)}$
$\displaystyle\leq C+C\|v(t)\|_{H^{1}(\Gamma)}^{q-2}\leq C.$

Using Hölder’s inequality, the continuous embedding
$H^{2}(\Omega)\hookrightarrow W^{1,6}(\Omega)$, interpolation inequality
(2.20), estimate (3.28), and Young’s inequality, we thus infer that for any
$\alpha\in(0,1)$ and almost all $t\in\mathbb{R}_{0}^{+}$,

$\displaystyle\big{\|}F^{\prime}\big{(}u(t)\big{)}\big{\|}_{H^{1}(\Omega)}^{2}=\big{\|}F^{\prime}\big{(}u(t)\big{)}\big{\|}_{L^{2}(\Omega)}^{2}+\big{\|}F^{\prime\prime}\big{(}u(t)\big{)}\nabla
u(t)\big{\|}_{L^{2}(\Omega)}^{2}$ $\displaystyle\leq
C+C\big{\|}F^{\prime\prime}\big{(}u(t)\big{)}\big{\|}_{L^{3}(\Omega)}^{2}\big{\|}\nabla
u(t)\big{\|}_{L^{6}(\Omega)}^{2}\leq
C+C\big{\|}u(t)\big{\|}_{H^{1}(\Omega)}\,\big{\|}u(t)\big{\|}_{H^{3}(\Omega)}$
$\displaystyle\leq\frac{C}{\alpha}+\alpha\big{\|}u(t)\big{\|}_{H^{3}(\Omega)}^{2}.$
(3.31)

Proceeding similarly, we deduce that for any $\alpha\in(0,1)$ and for any
$\alpha>0$ and almost all $t\in\mathbb{R}_{0}^{+}$,

$\displaystyle\big{\|}G^{\prime}\big{(}v(t)\big{)}\big{\|}_{H^{1}(\Gamma)}^{2}\leq\frac{C}{\alpha}+\alpha\big{\|}v(t)\big{\|}_{H^{3}(\Gamma)}^{2}.$
(3.32)

Furthermore, a regularity estimate for elliptic problems with bulk-surface
coupling (see [43, Thm. 3.3]) implies that for almost all
$t\in\mathbb{R}_{0}^{+}$,

$\displaystyle\big{\|}\big{(}u(t),v(t)\big{)}\big{\|}_{\mathcal{H}^{3}}^{2}$
$\displaystyle\leq
C\big{\|}\mu(t)-F^{\prime}\big{(}u(t)\big{)}\big{\|}_{H^{1}(\Omega)}^{2}+C\big{\|}\theta(t)-G^{\prime}\big{(}u(t)\big{)}\big{\|}_{H^{1}(\Gamma)}^{2}$
$\displaystyle\leq
C\big{\|}\big{(}\mu(t),\theta(t)\big{)}\big{\|}_{\mathcal{H}^{1}}^{2}+C\big{\|}F^{\prime}\big{(}u(t)\big{)}\big{\|}_{H^{1}(\Omega)}^{2}+C\big{\|}G^{\prime}\big{(}v(t)\big{)}\big{\|}_{H^{1}(\Gamma)}^{2}.$

Eventually, invoking the estimates (3.27), (Proof of Theorem 3.1.) and (3.32),
and choosing $\alpha\in(0,1)$ sufficiently small, we conclude that for almost
all $t\in\mathbb{R}_{0}^{+}$,

$\displaystyle\big{\|}\big{(}u(t),v(t)\big{)}\big{\|}_{\mathcal{H}^{3}}^{2}\leq
C+C\big{\|}\big{(}\mu(t),\theta(t)\big{)}\big{\|}_{\mathcal{H}^{1}}^{2}\leq
C\left(\frac{t+1}{t}\right),$

which verifies the smoothing property (3.7). Thus, the proof of Theorem 3.1 is
complete. ∎

Using the interpolation inequalities from Section 2 and the energy inequality
(3.6), we further establish improved continuity results for the weak solution.

######  Proposition 3.3 (Improved continuity results).

Suppose that ($\mathrm{A1}$)–($\mathrm{A3}$) hold and that $L\in(0,\infty)$.
Let $m\in\mathbb{R}$ be arbitrary, let
$(u_{0},v_{0})\in{\mathcal{W}_{\beta,m}^{1}}$ be any initial datum, and let
$(u^{L},v^{L},\mu^{L},\theta^{L})$ denote the corresponding weak solution of
the system (3.1). Then it holds that

$\displaystyle(u^{L},v^{L})\in C([0,\infty);\mathcal{H}^{1})\cap
C((0,\infty);\mathcal{H}^{2}).$ (3.33)

  * Proof.

Step 1. We first prove that $(u^{L},v^{L})\in C((0,\infty);\mathcal{H}^{2})$.
Therefore, we fix an arbitrary time $t_{0}\in(0,\infty)$. Let further
$t\in(0,\infty)$ with $\frac{1}{2}t_{0}<t<t_{0}+1$ be arbitrary. We already
know from Theorem 3.1 that

$(u^{L},v^{L})\in
L^{\infty}\big{(}\tfrac{1}{2}t_{0},t_{0}+1;\mathcal{H}^{2}\big{)}\cap
C\big{(}[\tfrac{1}{2}t_{0},t_{0}+1];\mathcal{L}^{2}\big{)}.$

This entails that $(u^{L},v^{L})(t)\in\mathcal{H}^{2}$ for all
$t\in[\tfrac{1}{2}t_{0},t_{0}+1]$ (see, e.g., [57, Chap. III, §1, Lem. 1.4.]).
Using Lemma 2.4 and the smoothing property (3.7) we deduce that

$\displaystyle\big{\|}\big{(}u^{L}(t),v^{L}(t)\big{)}-\big{(}u^{L}(t_{0}),v^{L}(t_{0})\big{)}\big{\|}_{\mathcal{H}^{2}}^{2}$
$\displaystyle\quad\leq
C\big{\|}\big{(}u^{L}(t),v^{L}(t)\big{)}-\big{(}u^{L}(t_{0}),v^{L}(t_{0})\big{)}\big{\|}_{\mathcal{H}^{3}}^{\frac{4}{3}}\big{\|}\big{(}u^{L}(t),v^{L}(t)\big{)}-\big{(}u^{L}(t_{0}),v^{L}(t_{0})\big{)}\big{\|}_{\mathcal{L}^{2}}^{\frac{1}{3}}$
$\displaystyle\quad\leq
C\left[\left(\frac{2(2+t_{0})}{t_{0}}\right)^{\frac{1}{2}}+\left(\frac{1+t_{0}}{t_{0}}\right)^{\frac{1}{2}}\right]^{\frac{4}{3}}\big{\|}\big{(}u^{L}(t),v^{L}(t)\big{)}-\big{(}u^{L}(t_{0}),v^{L}(t_{0})\big{)}\big{\|}_{\mathcal{L}^{2}}^{\frac{1}{3}}.$

We now choose $T:=1+t_{0}$ and recall that $(u^{L},v^{L})\in
C([0,T];\mathcal{L}^{2})$. Hence, the last line of the above estimate tends to
zero as $t\to t_{0}$ which proves $(u^{L},v^{L})\in
C((0,\infty);\mathcal{H}^{2})$.

Step 2. We now want to show that $(u^{L},v^{L})\in
C([0,\infty);\mathcal{H}^{1})$. As $(u^{L},v^{L})\in
C((0,\infty);\mathcal{H}^{1})$ is already established, only the continuity at
$t=0$ needs to be investigated. To this end, let
$\\{t_{k}\\}_{k\in\mathbb{N}}\subset(0,\infty)$ be an arbitrary sequence with
$t_{k}\to 0$ as $k\to\infty$.

From the growth conditions on $F$ and $G$ in ($\mathrm{A3}$), we deduce that

$\displaystyle\big{\|}\big{(}u(t_{k}),v(t_{k})\big{)}\big{\|}_{\mathcal{H}^{1}}\leq
C+CE\big{(}u(t_{k}),v(t_{k})\big{)}\leq C+CE(u_{0},v_{0})\leq C$ (3.34)

for all $k\in\mathbb{N}$. Hence, there exists a subsequence
$\\{t_{k}^{*}\\}_{k\in\mathbb{N}}$ of $(t_{k})_{k\in\mathbb{N}}$ and a pair
$(u_{*},v_{*})\in\mathcal{H}^{1}$ such that

$\displaystyle\big{(}u^{L}(t_{k}^{*}),v^{L}(t_{k}^{*})\big{)}\rightharpoonup(u_{*},v_{*})\quad\text{in
$\mathcal{H}^{1}$.}$ (3.35)

Since $(u^{L},v^{L})\in C([0,\infty);\mathcal{L}^{2})$, we also know that

$\displaystyle\big{(}u^{L}(t_{k}),v^{L}(t_{k})\big{)}\to(u_{0},v_{0})\quad\text{in
$\mathcal{L}^{2}$}$ (3.36)

and thus $(u_{*},v_{*})=(u_{0},v_{0})$ a.e. in $\Omega\times\Gamma$ due to
uniqueness of the limit. This means that the weak limit in (3.35) does not
depend on the subsequence extraction and thus, (3.35) holds true for the
_whole_ sequence, that is

$\displaystyle\big{(}u^{L}(t_{k}),v^{L}(t_{k})\big{)}\rightharpoonup(u_{0},v_{0})\quad\text{in
$\mathcal{H}^{1}$.}$ (3.37)

With some abuse of notation, we will write $E(t):=E\big{(}u(t),v(t)\big{)}$
for all $t\geq 0$. We further recall that $F_{1}$ and $G_{1}$ are convex, and
that $F_{2}$ and $G_{2}$ have at most quadratic growth. Using Lebesgue’s
general convergence theorem (see [2, Sect. 3.25]) for the non-convex
contributions, and lower-semicontinuity of the convex contributions, we obtain

$\displaystyle E(t_{0})\leq\underset{k\to\infty}{\lim\inf}\;E(t_{k}).$ (3.38)

Moreover, we infer from the energy inequality (3.6) that the sequence
$(E(t_{k}))_{k\in\mathbb{N}}$ is non-decreasing and bounded from above by
$E(t_{0})$. Hence, the limit $k\to\infty$ exists, and in combination with
(3.38) we conclude that

$\displaystyle E(t_{k})\to E(t_{0})\quad\text{as $k\to\infty$.}$ (3.39)

By the definition of the energy functional $E$ (see (3.2)), this implies that

$\displaystyle\begin{aligned} &\frac{1}{2}\big{\|}\big{(}\nabla
u^{L}(t_{k}),\nabla_{\Gamma}v^{L}(t_{k})\big{)}\big{\|}_{\mathcal{L}^{2}}^{2}-\frac{1}{2}\big{\|}\big{(}\nabla
u_{0},\nabla_{\Gamma}v_{0}\big{)}\big{\|}_{\mathcal{L}^{2}}^{2}\\\
&=E(t_{k})-E(t_{0})+\int_{\Omega}F\big{(}u_{0}\big{)}-F\big{(}u^{L}(t_{k})\big{)}\,\mathrm{d}x+\int_{\Gamma}G\big{(}v_{0}\big{)}-G\big{(}v^{L}(t_{k})\big{)}\,\mathrm{d}\Gamma.\end{aligned}$
(3.40)

Since $(u(t_{k}),v(t_{k}))\to(u_{0},v_{0})$ strongly in $\mathcal{L}^{2}$, and
$(u(t_{k}),v(t_{k}))_{k\in\mathbb{N}}$ is bounded in $\mathcal{H}^{1}$, we
also have $(u(t_{k}),v(t_{k}))\to(u_{0},v_{0})$ strongly in
$L^{p}(\Omega)\times L^{q}(\Gamma)$ via interpolation. We can thus use
Lebesgue’s general convergence theorem (see [2, Sect. 3.25]) to infer that the
right-hand side of (3.40) tends to zero as $k\to\infty$. Along with (3.36),
this yields

$\displaystyle\big{\|}\big{(}u^{L}(t_{k}),v^{L}(t_{k})\big{)}\big{\|}_{\mathcal{H}^{1}}^{2}-\big{\|}\big{(}u_{0},v_{0}\big{)}\big{\|}_{\mathcal{H}^{1}}^{2}\to
0\quad\text{$k\to\infty$.}$ (3.41)

Combining (3.37) and (3.41), we eventually conclude that
$(u^{L}(t_{k}),v^{L}(t_{k}))$ converges to $(u_{0},v_{0})$ strongly in
$\mathcal{H}^{1}$ as $k\to\infty$. As the null sequence
$(t_{k})_{k\in\mathbb{N}}$ was arbitrary, this proves $(u^{L},v^{L})\in
C([0,\infty);\mathcal{H}^{1})$.

Thus, the proof is complete. ∎

Moreover, we deduce continuous dependence of the weak solutions on the initial
data.

######  Proposition 3.4 (Continuous dependence on the initial data).

Suppose that ($\mathrm{A1}$)–($\mathrm{A3}$) hold and that $L\in(0,\infty)$.
Let $m\in\mathbb{R}$ be arbitrary. For $i\in\\{1,2\\}$, let
$(u_{0,i},v_{0,i})\in{\mathcal{W}_{\beta,m}^{1}}$ be any initial datum and let
$(u_{i}^{L},v_{i}^{L},\mu_{i}^{L},\theta_{i}^{L})$ denote the corresponding
weak solution of the system (3.1).

Then there exist positive, non-decreasing functions $\Lambda_{k}^{*}\in
C(\mathbb{R}_{0}^{+})$, $k\in\\{-1,1,2\\}$ depending only on $\Omega$,
$\beta$, $F$ and $G$ such that the following statements hold:

$\displaystyle\big{\|}\big{(}u^{L}_{2},v_{2}^{L}\big{)}(t)-\big{(}u^{L}_{1},v_{1}^{L}\big{)}(t)\big{\|}_{L,\beta,*}$
$\displaystyle\leq\Lambda_{-1}^{*}(t)\big{\|}(u_{0,2},v_{0,2})-(u_{0,1},v_{0,1})\big{\|}_{L,\beta,*},$
$\displaystyle\;\;t\geq 0,$ (3.42)
$\displaystyle\big{\|}\big{(}u^{L}_{2},v_{2}^{L}\big{)}(t)-\big{(}u^{L}_{1},v_{1}^{L}\big{)}(t)\big{\|}_{\mathcal{H}^{1}}$
$\displaystyle\leq\Lambda_{1}^{*}(t)\big{\|}(u_{0,2},v_{0,2})-(u_{0,1},v_{0,1})\big{\|}_{L,\beta,*},$
$\displaystyle\;\;t\geq 1.$ (3.43)
$\displaystyle\big{\|}\big{(}u^{L}_{2},v_{2}^{L}\big{)}(t)-\big{(}u^{L}_{1},v_{1}^{L}\big{)}(t)\big{\|}_{\mathcal{H}^{2}}$
$\displaystyle\leq\Lambda_{2}^{*}(t)\big{\|}(u_{0,2},v_{0,2})-(u_{0,1},v_{0,1})\big{\|}_{L,\beta,*}^{\frac{1}{2}},$
$\displaystyle\;\;t\geq 1.$ (3.44)

  * Proof.

Let $C$ denote generic positive constants depending only on $\Omega$, $\beta$,
$F$ and $G$. For brevity, we write

$\displaystyle(\bar{u},\bar{v},\bar{\mu},\bar{\theta}):=(u^{L}_{2},v^{L}_{2},\mu^{L}_{2},\theta^{L}_{2})-(u^{L}_{1},v^{L}_{1},\mu^{L}_{1},\theta^{L}_{1}),\quad(\bar{u}_{0},\bar{v}_{0})=(u_{0,2},v_{0,2})-(u_{0,1},v_{0,1}).$

From the weak formulation (3.4a)–(3.4d), we directly obtain that

$\displaystyle\langle\partial_{t}\bar{u}(t),w\rangle_{H^{1}(\Omega)}+\langle\partial_{t}\bar{v}(t),z\rangle_{H^{1}(\Gamma)}=\Big{(}\big{(}\bar{\mu}(t),\bar{\theta}(t)\big{)},(w,z)\Big{)}_{L,\beta}$
(3.45)

for almost all $t\in\mathbb{R}_{0}^{+}$ and all $(w,z)\in\mathcal{H}^{1}$, and

$\displaystyle\bar{\mu}(t)=-\Delta\bar{u}(t)+F^{\prime}\big{(}u_{2}^{L}(t)\big{)}-F^{\prime}\big{(}u_{1}^{L}(t)\big{)}$
$\displaystyle\quad\text{a.e.~{}in $\Omega$},$ (3.46)
$\displaystyle\bar{\theta}(t)=-\Delta_{\Gamma}\bar{v}(t)+G^{\prime}\big{(}v_{2}^{L}(t)\big{)}-G^{\prime}\big{(}v_{1}^{L}(t)\big{)}+\partial_{\mathbf{n}}\bar{u}(t)$
a.e. in $\Gamma$ (3.47)

for almost all $t\in\mathbb{R}_{0}^{+}$. Arguing similarly as in the proof of
Theorem 3.1 (cf. the derivation of (3.15)), we establish the estimate

$\displaystyle\dfrac{\mathrm{d}}{\mathrm{d}\tau}\big{\|}\big{(}\bar{u}(\tau),\bar{v}(\tau)\big{)}\big{\|}_{L,\beta,*}^{2}+\big{\|}\big{(}\nabla\bar{u}(\tau),\nabla_{\Gamma}\bar{v}(\tau)\big{)}\big{\|}_{\mathcal{L}^{2}}^{2}\leq
C\,\big{\|}\big{(}\bar{u}(\tau),\bar{v}(\tau)\big{)}\big{\|}_{L,\beta,*}^{2}$
(3.48)

for all $\tau\geq 0$. Hence, Gronwall’s lemma implies that

$\displaystyle\big{\|}\big{(}\bar{u}(t),\bar{v}(t)\big{)}\big{\|}_{L,\beta,*}^{2}+\int_{0}^{t}\big{\|}\big{(}\nabla\bar{u}(\tau),\nabla_{\Gamma}\bar{v}(\tau)\big{)}\big{\|}_{\mathcal{L}^{2}}^{2}\,\mathrm{d}s\leq
Ce^{Ct}\,\big{\|}\big{(}\bar{u}(0),\bar{v}(0)\big{)}\big{\|}_{L,\beta,*}^{2}$
(3.49)

for all $t\geq 0$. In particular, this proves (3.42).

Let now $T>1$ be arbitrary. Moreover, let $\varphi\in
C^{\infty}_{c}(\mathbb{R};\mathbb{R}_{0}^{+})$ with
$\mathrm{supp}\,\varphi\subset(-1,0)$ and $\|\varphi\|_{L^{1}(\mathbb{R})}=1$
be an arbitrary function. For any $k\in\mathbb{N}$, we set

$\displaystyle\varphi_{k}(s):=k\varphi(ks)\quad\text{for all
$s\in\mathbb{R}$.}$

This defines functions $\varphi_{k}\in
C^{\infty}_{c}(\mathbb{R};\mathbb{R}_{0}^{+})$ with
$\mathrm{supp}\,\varphi_{k}\subset(-\frac{1}{k},0)$ and
$\|\varphi_{k}\|_{L^{1}(\mathbb{R})}=1$ for all $k\in\mathbb{N}$. For any
Banach space $X$, any function $f\in L^{1}([0,T];X)$ and all $k\in\mathbb{N}$,
we now define

$\displaystyle[\hskip 1.0ptf\hskip 1.0pt]_{k}:\mathbb{R}_{0}^{+}\to
X,\quad[\hskip 1.0ptf\hskip
1.0pt]_{k}(t):=\int_{t}^{t+1}\varphi_{k}(t-s)\,f(s)\,\mathrm{d}s.$

It is well-known (cf. [2, Sect. 4.13]) that if $f\in L^{r}([0,T+1];X)$ for any
real number $r\in[1,\infty)$, then $[\hskip 1.0ptf\hskip 1.0pt]_{k}\in
C^{\infty}([0,T];X)\cap L^{r}([0,T];X)$ for all $k\in\mathbb{N}$, and $[\hskip
1.0ptf\hskip 1.0pt]_{k}\to f$ in $L^{r}([0,T];X)$ as $k\to\infty$.

Using this smooth approximation, we obtain

$\displaystyle\big{<}[\hskip 1.0pt\partial_{t}\bar{u}\hskip
1.0pt]_{k}(t),w\big{>}_{H^{1}(\Omega)}+\big{<}[\hskip
1.0pt\partial_{t}\bar{v}\hskip
1.0pt]_{k}(t),z\big{>}_{H^{1}(\Gamma)}=\Big{(}\big{(}[\hskip
1.0pt\bar{\mu}\hskip 1.0pt]_{k}(t),[\hskip 1.0pt\bar{\theta}\hskip
1.0pt]_{k}(t)\big{)},(w,z)\Big{)}_{L,\beta},$ (3.50)

for almost all $t\in(0,T)$, where

$\displaystyle[\hskip 1.0pt\bar{\mu}\hskip 1.0pt]_{k}$
$\displaystyle=-\Delta[\hskip 1.0pt\bar{u}\hskip
1.0pt]_{k}+\big{[}F^{\prime}(u_{2}^{L})-F^{\prime}(u_{1}^{L})\big{]}_{k}$
$\displaystyle\quad\text{a.e.~{}in $Q$},$ (3.51) $\displaystyle[\hskip
1.0pt\bar{\theta}\hskip 1.0pt]_{k}$ $\displaystyle=-\Delta_{\Gamma}[\hskip
1.0pt\bar{v}\hskip
1.0pt]_{k}+\big{[}G^{\prime}(v_{2}^{L})-G^{\prime}(v_{1}^{L})\big{]}_{k}+\partial_{\mathbf{n}}[\hskip
1.0pt\bar{u}\hskip 1.0pt]_{k}$ a.e. in $\Sigma$. (3.52)

Note that $\partial_{t}[\hskip 1.0pt\bar{u}\hskip 1.0pt]_{k}=[\hskip
1.0pt\partial_{t}\bar{u}\hskip 1.0pt]_{k}$ and $\partial_{t}[\hskip
1.0pt\bar{v}\hskip 1.0pt]_{k}=[\hskip 1.0pt\partial_{t}\bar{v}\hskip
1.0pt]_{k}$, which can be verified by using the change of variables $s\mapsto
t-s$ and Leibniz’s rule for differentiation under the integral sign. For more
details, see [2, Sect. 4.13].

Let us now choose an arbitrary number $t_{0}\in(0,1)$, and let $C_{0}$ denote
generic positive constants depending only on $\Omega$, $\beta$, $F$, $G$ and
$t_{0}$. We thus obtain

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}\tau}\Big{(}(\tau-
t_{0})\big{\|}\nabla[\hskip 1.0pt\bar{u}\hskip
1.0pt]_{k}(\tau)\big{\|}_{L^{2}(\Omega)}^{2}+(\tau-
t_{0})\big{\|}\nabla_{\Gamma}[\hskip 1.0pt\bar{v}\hskip
1.0pt]_{k}(\tau)\big{\|}_{L^{2}(\Gamma)}^{2}\Big{)}$
$\displaystyle=\frac{1}{2}\big{\|}\nabla[\hskip 1.0pt\bar{u}\hskip
1.0pt]_{k}(\tau)\big{\|}_{L^{2}(\Omega)}^{2}+\frac{1}{2}\big{\|}\nabla_{\Gamma}[\hskip
1.0pt\bar{v}\hskip 1.0pt]_{k}(\tau)\big{\|}_{L^{2}(\Gamma)}^{2}$
$\displaystyle\quad+(\tau-t_{0})\big{(}\nabla[\hskip 1.0pt\bar{u}\hskip
1.0pt]_{k}(\tau),\partial_{t}\nabla[\hskip 1.0pt\bar{u}\hskip
1.0pt]_{k}(\tau)\big{)}_{L^{2}(\Omega)}+(\tau-
t_{0})\big{(}\nabla_{\Gamma}[\hskip 1.0pt\bar{v}\hskip
1.0pt]_{k}(\tau),\partial_{t}\nabla_{\Gamma}[\hskip 1.0pt\bar{v}\hskip
1.0pt]_{k}(\tau)\big{)}_{L^{2}(\Gamma)}$
$\displaystyle=\frac{1}{2}\big{\|}\nabla[\hskip 1.0pt\bar{u}\hskip
1.0pt]_{k}(\tau)\big{\|}_{L^{2}(\Omega)}^{2}+\frac{1}{2}\big{\|}\nabla_{\Gamma}[\hskip
1.0pt\bar{v}\hskip 1.0pt]_{k}(\tau)\big{\|}_{L^{2}(\Gamma)}^{2}$
$\displaystyle\quad+(\tau-t_{0})\big{(}[\hskip 1.0pt\bar{\mu}\hskip
1.0pt]_{k}(\tau),\partial_{t}[\hskip 1.0pt\bar{u}\hskip
1.0pt]_{k}(\tau)\big{)}_{L^{2}(\Omega)}+(\tau-t_{0})\big{(}[\hskip
1.0pt\bar{\theta}\hskip 1.0pt]_{k}(\tau),\partial_{t}[\hskip
1.0pt\bar{v}\hskip 1.0pt]_{k}(\tau)\big{)}_{L^{2}(\Gamma)}$
$\displaystyle\quad-(\tau-
t_{0})\big{(}\big{[}F^{\prime}(u_{2}^{L})-F^{\prime}(u_{1}^{L})\big{]}_{k}(\tau),\partial_{t}[\hskip
1.0pt\bar{u}\hskip 1.0pt]_{k}(\tau)\big{)}_{L^{2}(\Omega)}$
$\displaystyle\quad-(\tau-
t_{0})\big{(}\big{[}G^{\prime}(v_{2}^{L})-G^{\prime}(v_{1}^{L})\big{]}_{k}(\tau),\partial_{t}[\hskip
1.0pt\bar{v}\hskip 1.0pt]_{k}(\tau)\big{)}_{L^{2}(\Gamma)}$
$\displaystyle=\frac{1}{2}\big{\|}\nabla[\hskip 1.0pt\bar{u}\hskip
1.0pt]_{k}(\tau)\big{\|}_{L^{2}(\Omega)}^{2}+\frac{1}{2}\big{\|}\nabla_{\Gamma}[\hskip
1.0pt\bar{v}\hskip 1.0pt]_{k}(\tau)\big{\|}_{L^{2}(\Gamma)}^{2}-(\tau-
t_{0})\big{\|}\big{(}\partial_{t}[\hskip 1.0pt\bar{u}\hskip
1.0pt]_{k}(\tau),\partial_{t}[\hskip 1.0pt\bar{v}\hskip
1.0pt]_{k}(\tau)\big{)}\big{\|}_{L,\beta,*}^{2}$ $\displaystyle\quad-(\tau-
t_{0})\big{(}\big{[}F^{\prime}(u_{2}^{L})-F^{\prime}(u_{1}^{L})\big{]}_{k}(\tau),\partial_{t}[\hskip
1.0pt\bar{u}\hskip 1.0pt]_{k}(\tau)\big{)}_{L^{2}(\Omega)}$
$\displaystyle\quad-(\tau-
t_{0})\big{(}\big{[}G^{\prime}(v_{2}^{L})-G^{\prime}(v_{1}^{L})\big{]}_{k}(\tau),\partial_{t}[\hskip
1.0pt\bar{v}\hskip 1.0pt]_{k}(\tau)\big{)}_{L^{2}(\Gamma)}$

for all $t_{0}<\tau<T$. Integrating with respect to $\tau$ from $t_{0}$ to an
arbitrary $t\in[t_{0},T]$, we infer that

$\displaystyle\begin{aligned}
&\frac{1}{2}\Big{(}(t-t_{0})\big{\|}\nabla[\hskip 1.0pt\bar{u}\hskip
1.0pt]_{k}(t)\big{\|}_{L^{2}(\Omega)}^{2}+(t-t_{0})\big{\|}\nabla_{\Gamma}[\hskip
1.0pt\bar{v}\hskip 1.0pt]_{k}(t)\big{\|}_{L^{2}(\Gamma)}^{2}\Big{)}\\\
&\quad+\int_{t_{0}}^{t}(\tau-t_{0})\big{\|}\big{(}\partial_{t}[\hskip
1.0pt\bar{u}\hskip 1.0pt]_{k}(\tau),\partial_{t}[\hskip 1.0pt\bar{v}\hskip
1.0pt]_{k}(\tau)\big{)}\big{\|}_{L,\beta,*}^{2}\,\mathrm{d}\tau\end{aligned}$
$\displaystyle\begin{aligned}
&=\frac{1}{2}\int_{t_{0}}^{t}\big{\|}\nabla[\hskip 1.0pt\bar{u}\hskip
1.0pt]_{k}(\tau)\big{\|}_{L^{2}(\Omega)}^{2}\,\mathrm{d}\tau+\frac{1}{2}\int_{t_{0}}^{t}\big{\|}\nabla_{\Gamma}[\hskip
1.0pt\bar{v}\hskip
1.0pt]_{k}(\tau)\big{\|}_{L^{2}(\Gamma)}^{2}\,\mathrm{d}\tau\\\
&\quad-\int_{t_{0}}^{t}(\tau-t_{0})\big{<}\partial_{t}[\hskip
1.0pt\bar{u}\hskip
1.0pt]_{k}(\tau),\big{[}F^{\prime}(u_{2}^{L})-F^{\prime}(u_{1}^{L})\big{]}_{k}(\tau)\big{>}_{H^{1}(\Omega)}\,\mathrm{d}\tau\\\
&\quad-\int_{t_{0}}^{t}(\tau-t_{0})\big{<}\partial_{t}[\hskip
1.0pt\bar{v}\hskip
1.0pt]_{k}(\tau),\big{[}G^{\prime}(v_{2}^{L})-G^{\prime}(v_{1}^{L})\big{]}_{k}(\tau)\big{>}_{H^{1}(\Gamma)}\,\mathrm{d}\tau\end{aligned}$

for all $t\in[t_{0},T]$. Passing to the limit $k\to\infty$, we obtain

$\displaystyle\frac{1}{2}\Big{(}(t-t_{0})\big{\|}\nabla\bar{u}(t)\big{\|}_{L^{2}(\Omega)}^{2}+(t-t_{0})\big{\|}\nabla_{\Gamma}\bar{v}(t)\big{\|}_{L^{2}(\Gamma)}^{2}\Big{)}$
$\displaystyle\;\;+\int_{t_{0}}^{t}(\tau-
t_{0})\big{\|}\big{(}\partial_{t}\bar{u},\partial_{t}\bar{v}\big{)}(\tau)\big{\|}_{L,\beta,*}^{2}\,\mathrm{d}\tau$
$\displaystyle=\frac{1}{2}\int_{t_{0}}^{t}\big{\|}\nabla\bar{u}(\tau)\big{\|}_{L^{2}(\Omega)}^{2}\,\mathrm{d}\tau+\frac{1}{2}\int_{t_{0}}^{t}\big{\|}\nabla_{\Gamma}\bar{v}(\tau)\big{\|}_{L^{2}(\Gamma)}^{2}\,\mathrm{d}\tau$
$\displaystyle\;\;-\int_{t_{0}}^{t}(\tau-
t_{0})\big{<}\partial_{t}\bar{u}(\tau),F^{\prime}\big{(}u_{2}^{L}(\tau)\big{)}-F^{\prime}(u_{1}^{L}\big{(}\tau)\big{)}\big{>}_{H^{1}(\Omega)}\,\mathrm{d}\tau$
$\displaystyle\;\;-\int_{t_{0}}^{t}(\tau-
t_{0})\big{<}\partial_{t}\bar{v}(\tau),G^{\prime}\big{(}v_{2}^{L}(\tau)\big{)}-G^{\prime}\big{(}v_{1}^{L}(\tau)\big{)}\big{>}_{H^{1}(\Gamma)}\,\mathrm{d}\tau$
$\displaystyle=\frac{1}{2}\int_{t_{0}}^{t}\big{\|}\nabla\bar{u}(\tau)\big{\|}_{L^{2}(\Omega)}^{2}\,\mathrm{d}\tau+\frac{1}{2}\int_{t_{0}}^{t}\big{\|}\nabla_{\Gamma}\bar{v}(\tau)\big{\|}_{L^{2}(\Gamma)}^{2}\,\mathrm{d}\tau$
$\displaystyle\;\;+\int_{t_{0}}^{t}(\tau-
t_{0})\Big{(}\SS^{L}\big{(}(\partial_{t}\bar{u},\partial_{t}\bar{v})(\tau)\big{)},\big{[}F^{\prime}\big{(}u_{2}^{L}(\tau)\big{)}-F^{\prime}\big{(}u_{1}^{L}(\tau)\big{)},G^{\prime}\big{(}v_{2}^{L}(\tau)\big{)}-G^{\prime}\big{(}v_{1}^{L}(\tau)\big{)}\big{]}\Big{)}_{L,\beta}\mathrm{d}\tau$
$\displaystyle\leq\frac{1}{2}\int_{t_{0}}^{t}\big{\|}\nabla\bar{u}(\tau)\big{\|}_{L^{2}(\Omega)}^{2}\,\mathrm{d}\tau+\frac{1}{2}\int_{t_{0}}^{t}\big{\|}\nabla_{\Gamma}\bar{v}(\tau)\big{\|}_{L^{2}(\Gamma)}^{2}\,\mathrm{d}\tau$
$\displaystyle\;\;+C\int_{t_{0}}^{t}(\tau-
t_{0})\Big{(}\big{\|}F^{\prime}\big{(}u_{2}^{L}(\tau)\big{)}-F^{\prime}\big{(}u_{1}^{L}(\tau)\big{)}\big{\|}_{H^{1}(\Omega)}^{2}+\big{\|}G^{\prime}\big{(}v_{2}^{L}(\tau)\big{)}-G^{\prime}\big{(}v_{1}^{L}(\tau)\big{)}\big{\|}_{H^{1}(\Gamma)}^{2}\Big{)}\,\mathrm{d}\tau$
$\displaystyle\;\;+\frac{1}{2}\int_{t_{0}}^{t}(\tau-
t_{0})\big{\|}\big{(}\partial_{t}\bar{u},\partial_{t}\bar{v}\big{)}(\tau)\big{\|}_{L,\beta,*}^{2}\,\mathrm{d}\tau$
(3.53)

for all $t\in[t_{0},T]$. Since $T>0$ was arbitrary, (Proof.) actually holds
true for all $t\geq t_{0}$. Invoking the smoothing property (3.7) and
Sobolev’s embedding theorem, we infer that

$\displaystyle\big{\|}u_{i}^{L}(\tau)\big{\|}_{W^{1,\infty}(\Omega)}+\big{\|}v_{i}^{L}(\tau)\big{\|}_{W^{1,\infty}(\Gamma)}\leq
C\big{\|}\big{(}u_{i}^{L}(\tau),v_{i}^{L}(\tau)\big{)}\big{\|}_{\mathcal{H}^{3}}\leq
C\left(\dfrac{1+t_{0}}{t_{0}}\right)^{\frac{1}{2}}$ (3.54)

for almost all $\tau\geq t_{0}$. By a straightforward computation, this
entails that

$\displaystyle\big{\|}F^{\prime}\big{(}u_{2}^{L}(\tau)\big{)}-F^{\prime}\big{(}u_{1}^{L}(\tau)\big{)}\big{\|}_{H^{1}(\Omega)}^{2}$
$\displaystyle\leq C_{0}\|\bar{u}(\tau)\|_{H^{1}(\Omega)}^{2},$ (3.55)
$\displaystyle\big{\|}G^{\prime}\big{(}v_{2}^{L}(\tau)\big{)}-G^{\prime}\big{(}v_{1}^{L}(\tau)\big{)}\big{\|}_{H^{1}(\Gamma)}^{2}$
$\displaystyle\leq C_{0}\|\bar{v}(\tau)\|_{H^{1}(\Gamma)}^{2},$ (3.56)

for almost all $\tau\geq t_{0}$ and $i\in\\{1,2\\}$. Plugging the estimates
(3.55) and (3.56) into (Proof.), and applying Lemma 2.3, we deduce that

$\displaystyle(t-t_{0})\big{\|}\big{(}\nabla\bar{u},\nabla_{\Gamma}\bar{v}\big{)}(t)\big{\|}_{\mathcal{L}^{2}}^{2}$
$\displaystyle\quad\leq
C_{0}\int_{0}^{t}\big{\|}\big{(}\nabla\bar{u},\nabla_{\Gamma}\bar{v}\big{)}(\tau)\big{\|}_{\mathcal{L}^{2}}^{2}\,\mathrm{d}\tau+C_{0}\int_{t_{0}}^{t}(\tau-
t_{0})\,\big{\|}\big{(}\nabla\bar{u},\nabla_{\Gamma}\bar{v}\big{)}(\tau)\big{\|}_{\mathcal{L}^{2}}^{2}\,\mathrm{d}\tau$
$\displaystyle\qquad+C_{0}\int_{t_{0}}^{t}(\tau-
t_{0})\,\big{\|}\big{(}\bar{u},\bar{v}\big{)}(\tau)\big{\|}_{L,\beta,*}^{2}\,\mathrm{d}\tau$
(3.57)

for all $t\geq t_{0}$. Recalling (3.49), we infer that

$\displaystyle(t-t_{0})\big{\|}\big{(}\nabla\bar{u},\nabla_{\Gamma}\bar{v}\big{)}(t)\big{\|}_{\mathcal{L}^{2}}^{2}$
$\displaystyle\quad\leq C_{0}\int_{t_{0}}^{t}(\tau-
t_{0})\,\big{\|}\big{(}\nabla\bar{u},\nabla_{\Gamma}\bar{v}\big{)}(\tau)\big{\|}_{\mathcal{L}^{2}}^{2}\,\mathrm{d}\tau+C_{0}\,e^{Ct}\,\big{(}1+(t-t_{0})^{2}\big{)}\,\big{\|}\big{(}\bar{u},\bar{v}\big{)}(0)\big{\|}_{L,\beta,*}^{2}$
(3.58)

for all $t\geq t_{0}$. We now fix $t_{0}:=\tfrac{1}{2}$. Hence, Gronwall’s
lemma eventually implies that

$\displaystyle\big{\|}\big{(}\nabla\bar{u},\nabla_{\Gamma}\bar{v}\big{)}(t)\big{\|}_{\mathcal{L}^{2}}^{2}\leq
C\,e^{Ct}\,\frac{\big{(}1+(t-\tfrac{1}{2})^{2}\big{)}}{t-\tfrac{1}{2}}\,\big{\|}\big{(}\bar{u},\bar{v}\big{)}(0)\big{\|}_{L,\beta,*}^{2}$
(3.59)

for all $t\geq\tfrac{1}{2}$. In view of (3.42) and Lemma 2.3, the compact
Lipschitz estimate (3.43) now directly follows.

Using Lemma 2.4, the smoothing property (3.7) and the estimate (3.43), we
further conclude that

$\displaystyle\big{\|}\big{(}\bar{u}(t),\bar{v}(t)\big{)}\big{\|}_{\mathcal{H}^{2}}^{2}\leq
C\,\big{\|}\big{(}\bar{u}(t),\bar{v}(t)\big{)}\big{\|}_{\mathcal{H}^{3}}\big{\|}\big{(}\bar{u}(t),\bar{v}(t)\big{)}\big{\|}_{\mathcal{H}^{1}}\leq
C\,\Lambda_{1}^{*}(t)\big{\|}\big{(}\bar{u}_{0},\bar{v}_{0}\big{)}\big{\|}_{L,\beta,*}$

for almost all $t\geq 1$. This proves (3.44) and thus, the proof is complete.
∎

We can further show that the weak solution is Hölder continuous in time with a
Hölder constant being uniform in $L$.

######  Proposition 3.5 (Uniform Hölder estimate).

Suppose that ($\mathrm{A1}$)–($\mathrm{A3}$) hold and that $L\in(0,\infty)$.
Let $m\in\mathbb{R}$ be arbitrary, let
$(u_{0},v_{0})\in{\mathcal{W}_{\beta,m}^{1}}$ be any initial datum, and let
$(u^{L},v^{L},\mu^{L},\theta^{L})$ denote the corresponding weak solution of
the system (3.1).

Then there exists a positive constant $\Theta^{*}$ depending only on $\Omega$
and $E(u_{0},v_{0})$ such that

$\displaystyle\big{\|}(u^{L},v^{L})(t)-(u^{L},v^{L})(s)\big{\|}_{\mathcal{H}^{1}}\leq\Theta^{*}\,\left|s-t\right|^{\frac{3}{16}},\quad\text{for
all $s,t\geq 1$.}$ (3.60)

  * Proof.

Let $C$ denote a generic positive constant depending only on $\Omega$ and
$E(u_{0},v_{0})$ that may change its value from line to line. Let $s,t\geq 0$
be arbitrary, and without loss of generality we assume $t>s$. Moreover, let
$w\in H^{1}_{0}(\Omega)$ (i.e., $w|_{\Gamma}=0$) be an arbitrary test
function. Integrating (3.4a) with respect to time, we get

$\displaystyle\big{<}u^{L}(t)-u^{L}(s),w\big{>}_{H^{1}(\Omega)}=-\int_{s}^{t}\nabla\mu^{L}(\tau)\cdot\nabla
w\,\mathrm{d}\tau.$

Now, taking the supremum over all test functions $w\in H^{1}_{0}(\Omega)$ with
$\|w\|_{H^{1}(\Omega)}=1$, and recalling the energy inequality (3.6), we infer
that

$\displaystyle\big{\|}u^{L}(t)-u^{L}(s)\big{\|}_{H^{-1}(\Omega)}\leq\int_{s}^{t}\big{\|}\nabla\mu^{L}(\tau)\big{\|}_{L^{2}(\Omega)}\,\mathrm{d}\tau$
$\displaystyle\quad\leq\left|t-s\right|^{\frac{1}{2}}\left(\int_{0}^{t}\big{\|}\nabla\mu^{L}(\tau)\big{\|}_{L^{2}(\Omega)}^{2}\,\mathrm{d}\tau\right)^{\\!\frac{1}{2}}\leq\sqrt{2E(u_{0},v_{0})}\,\left|t-s\right|^{\frac{1}{2}},$

where $H^{-1}(\Omega)$ stands for the dual space of $H^{1}_{0}(\Omega)$. We
now suppose that $s,t\geq 1$. Interpolating $H^{3/2}(\Omega)$ between
$H^{-1}(\Omega)$ and $H^{3}(\Omega)$ (see, e.g., [58, Sec. 4.3.1]), and
invoking the smoothing property (3.7), we obtain

$\displaystyle\big{\|}u^{L}(t)-u^{L}(s)\big{\|}_{H^{3/2}(\Omega)}$
$\displaystyle\leq
C\big{\|}u^{L}(t)-u^{L}(s)\big{\|}_{H^{-1}(\Omega)}^{\frac{3}{8}}\,\big{\|}u^{L}(t)-u^{L}(s)\big{\|}_{H^{3}(\Omega)}^{\frac{5}{8}}$
$\displaystyle\leq C\left|t-s\right|^{\frac{3}{16}}.$

Eventually, by means of the continuous embedding
$H^{3/2}(\Omega)\hookrightarrow H^{1}(\Gamma)$, we conclude that

$\displaystyle\big{\|}(u^{L},v^{L})(t)-(u^{L},v^{L})(s)\big{\|}_{\mathcal{H}^{1}}\leq
C\left|t-s\right|^{\frac{3}{16}},$

which completes the proof. ∎

As a consequence of Theorem 3.1, Proposition 3.3 and Proposition 3.4, the weak
solutions to the problem (3.1) can be described via a strongly continuous
semigroup.

######  Corollary 3.6.

Suppose that ($\mathrm{A1}$)–($\mathrm{A3}$) hold and that $L\in(0,\infty)$.
Let $m\in\mathbb{R}$ be arbitrary. For any
$(u_{0},v_{0})\in{\mathcal{W}_{\beta,m}^{1}}$ let
$(u^{L},v^{L},\mu^{L},\theta^{L})$ denote the corresponding unique weak
solution of the system (3.1).

For any $t\geq 0$, we define the operator

$\displaystyle
S_{m}^{L}(t):{\mathcal{W}_{\beta,m}^{1}}\to{\mathcal{W}_{\beta,m}^{1}},\quad(u_{0},v_{0})\mapsto\big{(}u^{L}(t),v^{L}(t)\big{)}.$
(3.61)

Then the family $\big{\\{}S_{m}^{L}(t)\big{\\}}_{t\geq 0}$ defines a strongly
continuous semigroup on ${\mathcal{W}_{\beta,m}^{1}}$.

###  3.2 The GMS model ($L=0$)

######  Theorem 3.7 (Strong well-posedness and smoothing property for the GMS
model).

Suppose that ($\mathrm{A1}$)–($\mathrm{A3}$) hold and that $L=0$. Let
$m\in\mathbb{R}$ be arbitrary and let
$(u_{0},v_{0})\in{\mathcal{W}_{\beta,m}^{1}}$ be any initial datum.

Then there exists a unique global weak solution
$(u^{0},v^{0},\mu^{0},\theta^{0})$ of the system (3.1) existing on the time
interval $\mathbb{R}_{0}^{+}$ and having the following properties:

  1. $(\mathrm{i})$

For every $T>0$, the solution has the regularity

$\displaystyle\begin{aligned} \big{(}u^{0},v^{0}\big{)}&\in
C^{0,\frac{1}{4}}([0,T];\mathcal{L}^{2})\cap L^{\infty}(0,T;{\mathcal{V}})\\\
&\qquad\cap L^{2}(0,T;\mathcal{H}^{3})\cap
H^{1}\big{(}0,T;({\mathcal{D}_{\beta}})^{\prime}\big{)},\end{aligned}$ (3.62a)
$\displaystyle\big{(}\mu^{0},\theta^{0}\big{)}\in L^{2}(0,T;\mathcal{H}^{1}).$
(3.62b)

  2. $(\mathrm{ii})$

The solution satisfies

$\displaystyle\big{<}(\partial_{t}u^{0},\partial_{t}v^{0}),(w,z)\big{>}_{{\mathcal{D}_{\beta}}}$
$\displaystyle=-\int_{\Omega}\nabla\mu^{0}\cdot\nabla
w\,\mathrm{d}x-\int_{\Gamma}\nabla_{\Gamma}\theta^{0}\cdot\nabla_{\Gamma}z\,\mathrm{d}\Gamma$
(3.63a) a.e. on $\mathbb{R}_{0}^{+}$ for all test functions
$w\in{\mathcal{D}_{\beta}}$, and $\displaystyle\mu^{0}=-\Delta
u^{0}+F^{\prime}(u^{0})$ $\displaystyle\quad\text{a.e.~{}in Q},$ (3.63b)
$\displaystyle\theta^{0}=-\Delta_{\Gamma}v^{0}+G^{\prime}(v^{0})+\partial_{\mathbf{n}}u^{0}$
$\displaystyle\quad\text{a.e.~{}in $\Sigma$},$ (3.63c)
$\displaystyle\mu^{0}|_{{\Sigma}}=\beta\theta^{0},\quad
u^{0}|_{{\Sigma}}=v^{0}$ $\displaystyle\quad\text{a.e.~{}in $\Sigma$},$
(3.63d) $\displaystyle(u^{0},v^{0})|_{t=0}=(u_{0},v_{0})$
$\displaystyle\quad\text{a.e.~{}in }\Omega\times\Gamma.$ (3.63e)

  3. $(\mathrm{iii})$

The solution satisfies the mass conservation law

$\displaystyle\beta\int_{\Omega}u^{0}(t)\,\mathrm{d}x+\int_{\Gamma}v^{0}(t)\,\mathrm{d}\Gamma=\beta\int_{\Omega}u_{0}\,\mathrm{d}x+\int_{\Gamma}v_{0}\,\mathrm{d}\Gamma=m,\quad$
(3.64)

for all $t\in\mathbb{R}_{0}^{+}$, i.e.,
$\big{(}u^{0}(t),v^{0}(t)\big{)}\in{\mathcal{W}_{\beta,m}^{1}}$ for almost all
$t\in\mathbb{R}_{0}^{+}$.

  4. $(\mathrm{iv})$

The solution satisfies the energy inequality

$\displaystyle
E\big{(}u^{0}(t),v^{0}(t)\big{)}+\frac{1}{2}\int_{0}^{t}\big{\|}\big{(}\mu^{0}(s),\theta^{0}(s)\big{)}\big{\|}_{0,\beta}^{2}\leq
E(u_{0},v_{0})$ (3.65)

for almost all $t\geq 0$.

  5. $(\mathrm{v})$

The solution satisfies the smoothing property, i.e. there exists a constant
$C_{0}>0$ depending only on $\Omega$, $\beta$, $F$, $G$ and
$\|(u_{0},v_{0})\|_{\mathcal{H}^{1}}$, such that

$\displaystyle\big{\|}\big{(}u^{0}(t),v^{0}(t)\big{)}\big{\|}_{\mathcal{H}^{3}}\leq
C_{0}\left(\dfrac{1+t}{t}\right)^{\frac{1}{2}}$ (3.66)

for almost all $t\geq 0$.

  * Proof.

For the existence and uniqueness of a weak solution satisfying the assertions
(i)–(iv), we refer the reader to [42, Prop. 3.4].

As the proof of the smoothing property (v) is very similar to the approach in
the proof of Theorem 3.1, we only sketch the most important steps. To provide
a cleaner presentation we omit the superscript $0$. Let
$(w,z)\in{\mathcal{D}_{\beta}}$ be arbitrary (i.e., $w|_{\Gamma}=\beta z$ a.e.
on $\Gamma$). Testing (3.63a) with $(w,z)$ and recalling the definition of the
product $\langle\cdot,\cdot\rangle_{{\mathcal{D}_{\beta}}}$ (see
($\mathrm{P4}$)), we obtain

$\displaystyle\begin{aligned}
\big{\langle}(\partial_{t}u,\partial_{t}v),(w,z)\big{\rangle}_{{\mathcal{D}_{\beta}}}&=-\big{(}\nabla\mu,\nabla
w\big{)}_{L^{2}(\Omega)}-\big{(}\nabla_{\Gamma}\theta,\nabla_{\Gamma}z\big{)}_{L^{2}(\Gamma)}\\\
&=-\big{(}(\mu,\theta),(w,z)\big{)}_{0,\beta}.\end{aligned}$ (3.67)

Hence, using the notation for the difference quotient introduced in (3.10), we
infer that

$\displaystyle\begin{aligned}
\big{\langle}(\partial_{t}^{h}\partial_{t}u(\tau),\partial_{t}^{h}\partial_{t}v(\tau)),(w,z)\big{\rangle}_{{\mathcal{D}_{\beta}}}=-\Big{(}\big{(}\partial_{t}^{h}\mu(\tau),\partial_{t}^{h}\theta(\tau)\big{)},(w,z)\Big{)}_{0,\beta}\end{aligned}$
(3.68)

for almost all $\tau\in\mathbb{R}_{0}^{+}$, where $\partial_{t}^{h}\mu(\tau)$
and $\partial_{t}^{h}\theta(\tau)$ can be expressed by means of
$\eqref{WF:GMS:3}$ and $\eqref{WF:GMS:4}$. Choosing the test function
$(w,z)=\SS^{0}\big{(}\partial_{t}^{h}u(\tau),\partial_{t}^{h}v(\tau)\big{)}$,
a straightforward computation reveals that

$\displaystyle-\dfrac{1}{2}\dfrac{\mathrm{d}}{\mathrm{d}\tau}\big{\|}\partial_{t}^{h}u(\tau)\big{\|}_{0,\beta,*}^{2}$
$\displaystyle\quad=\big{\|}\nabla\partial_{t}^{h}u(\tau)\big{\|}^{2}_{L^{2}(\Omega)}+\dfrac{1}{h}\int_{\Omega}\Big{(}F^{\prime}\big{(}u(\tau+h)\big{)}-F^{\prime}\big{(}u(\tau)\big{)}\Big{)}\partial_{t}^{h}u(\tau)\,\mathrm{d}x$
$\displaystyle\qquad+\big{\|}\nabla_{\Gamma}\partial_{t}^{h}v(\tau)\big{\|}^{2}_{L^{2}(\Gamma)}+\dfrac{1}{h}\int_{\Gamma}\Big{(}G^{\prime}\big{(}v(\tau+h)\big{)}-G^{\prime}\big{(}v(\tau)\big{)}\Big{)}\partial_{t}^{h}v(\tau)\,\mathrm{d}\Gamma$

for almost all $\tau\in\mathbb{R}_{0}^{+}$. Based on this identity, the
smoothing property can be established by proceeding exactly as in the proof of
Theorem 3.1. Note that the regularity estimates in [43, Thm. 3.3] are also
applicable for bulk-surface elliptic problems with Dirichlet type coupling
conditions. ∎

As for the KLLM model, we obtain an improved continuity result,
Lipschitz/Hölder continuous dependence of the weak solutions on the initial
data, and a uniform Hölder estimate for times in $[1,\infty)$.

######  Proposition 3.8 (Improved continuity results).

Suppose that ($\mathrm{A1}$)–($\mathrm{A3}$) hold and that $L=0$. Let
$m\in\mathbb{R}$ be arbitrary, let
$(u_{0},v_{0})\in{\mathcal{W}_{\beta,m}^{1}}$ be any initial datum, and let
$(u^{0},v^{0},\mu^{0},\theta^{0})$ denote the corresponding weak solution of
the system (3.1). Then it holds that

$\displaystyle(u^{0},v^{0})\in C([0,\infty);\mathcal{H}^{1})\cap
C((0,\infty);\mathcal{H}^{2}).$ (3.69)

######  Proposition 3.9 (Continuous dependence on the initial data).

Suppose that ($\mathrm{A1}$)–($\mathrm{A3}$) hold and that $L=0$. Let
$m\in\mathbb{R}$ be arbitrary. For $i\in\\{1,2\\}$, let
$(u_{0,i},v_{0,i})\in{\mathcal{W}_{\beta,m}^{1}}$ be any initial datum and let
$(u_{i}^{0},v_{i}^{0},\mu_{i}^{0},\theta_{i}^{0})$ denote the corresponding
weak solution of the system (3.1).

Then there exist positive, non-decreasing functions $\Lambda_{k}^{0}\in
C(\mathbb{R}_{0}^{+})$, $k\in\\{-1,1,2\\}$ depending only on $\Omega$,
$\beta$, $F$ and $G$ such that the following statements hold:

$\displaystyle\big{\|}\big{(}u^{0}_{2},v_{2}^{0}\big{)}(t)-\big{(}u^{0}_{1},v_{1}^{0}\big{)}(t)\big{\|}_{0,\beta,*}$
$\displaystyle\leq\Lambda_{-1}^{0}(t)\big{\|}(u_{0,2},v_{0,2})-(u_{0,1},v_{0,1})\big{\|}_{0,\beta,*},$
$\displaystyle\;\;t\geq 0,$ (3.70)
$\displaystyle\big{\|}\big{(}u^{0}_{2},v_{2}^{0}\big{)}(t)-\big{(}u^{0}_{1},v_{1}^{0}\big{)}(t)\big{\|}_{\mathcal{H}^{1}}$
$\displaystyle\leq\Lambda_{1}^{0}(t)\big{\|}(u_{0,2},v_{0,2})-(u_{0,1},v_{0,1})\big{\|}_{0,\beta,*},$
$\displaystyle\;\;t\geq 1,$ (3.71)
$\displaystyle\big{\|}\big{(}u^{0}_{2},v_{2}^{0}\big{)}(t)-\big{(}u^{0}_{1},v_{1}^{0}\big{)}(t)\big{\|}_{\mathcal{H}^{2}}$
$\displaystyle\leq\Lambda_{2}^{0}(t)\big{\|}(u_{0,2},v_{0,2})-(u_{0,1},v_{0,1})\big{\|}_{0,\beta,*}^{\frac{1}{2}},$
$\displaystyle\;\;t\geq 1.$ (3.72)

######  Proposition 3.10 (Uniform Hölder estimate).

Suppose that ($\mathrm{A1}$)–($\mathrm{A3}$) hold and that $L=0$. Let
$m\in\mathbb{R}$ be arbitrary, let
$(u_{0},v_{0})\in{\mathcal{W}_{\beta,m}^{1}}$ be any initial datum, and let
$(u^{0},v^{0},\mu^{0},\theta^{0})$ denote the corresponding weak solution of
the system (3.1).

Then there exists a positive constant $\Theta^{0}$ depending only on $\Omega$
and $E(u_{0},v_{0})$ such that

$\displaystyle\big{\|}(u^{0},v^{0})(t)-(u^{0},v^{0})(s)\big{\|}_{\mathcal{H}^{1}}$
$\displaystyle\leq\Theta^{0}\,\left|s-t\right|^{\frac{3}{16}},$
$\displaystyle\quad t\geq 1.$ (3.73)

The proofs of Proposition 3.8, Proposition 3.10 and Proposition 3.9 are very
similar to those of the corresponding results for the KLLM model. Therefore,
they will not be presented.

As a consequence of Theorem 3.7, Proposition 3.8 and Proposition 3.9, the weak
solutions to the problem (3.1) can be described via a strongly continuous
semigroup.

######  Corollary 3.11.

Suppose that ($\mathrm{A1}$)–($\mathrm{A3}$) hold and that $L=0$. Let
$m\in\mathbb{R}$ be arbitrary. For any
$(u_{0},v_{0})\in{\mathcal{W}_{\beta,m}^{1}}$ let
$(u^{0},v^{0},\mu^{0},\theta^{0})$ denote the corresponding unique weak
solution.

For any $t\geq 0$, we define the operator

$\displaystyle
S_{m}^{0}:{\mathcal{W}_{\beta,m}^{1}}\to{\mathcal{W}_{\beta,m}^{1}},\quad(u_{0},v_{0})\mapsto\big{(}u^{0}(t),v^{0}(t)\big{)}.$
(3.74)

Then the family $\big{\\{}S_{m}^{0}(t)\big{\\}}_{t\geq 0}$ defines a strongly
continuous semigroup on ${\mathcal{W}_{\beta,m}^{1}}$.

##  4 Long-time dynamics and existence of a global attractor

###  4.1 Long-time dynamics of the KLLM model

In this section, we investigate the long-time behavior of the KLLM model (3.1)
for any fixed $L\in(0,\infty)$. We prove the existence of a global attractor
for weak solutions of this system, and we establish the convergence of each
weak solution to a single stationary point as $t\rightarrow\infty$. To this
end, we first analyze the set of stationary points of the problem (3.1) with
$L\in(0,\infty)$.

####  4.1.1 The set of stationary points

For the semigroup ${\left\\{S_{m}^{L}(t)\right\\}}_{t\geq 0}$ on
${\mathcal{W}_{\beta,m}^{1}}$ corresponding to the KLLM model with fixed
$L\in(0,\infty)$, the corresponding set of stationary points of the problem
(3.1) can be expressed as follows:

$\displaystyle\mathcal{N}_{m}^{L}=\left\\{(u,v)\in{\mathcal{W}_{\beta,m}^{1}}:S_{m}^{L}(t)(u,v)=(u,v)\;\text{for
all}\;t\geq 0\right\\}.$

We aim to show that the set $\mathcal{N}_{m}^{L}$ is a nonempty and bounded
subset of ${\mathcal{W}_{\beta,m}^{2}}$.

To this end, let $(u,v)\in\mathcal{N}_{m}^{L}$ be arbitrary, and let $\mu$ and
$\theta$ denote the corresponding chemical potentials given by (3.4c) and
(3.4d). Then the energy inequality (3.6) entails that

$\int_{0}^{T}\left(\left\|\nabla\mu\right\|_{L^{2}\left(\Omega\right)}^{2}+\left\|\nabla\theta\right\|_{L^{2}\left(\Gamma\right)}^{2}+\dfrac{1}{L}\left\|\beta\theta-\mu\right\|_{L^{2}\left(\Gamma\right)}^{2}\right)dt\leq
0.$

Hence, there exists a constant $\lambda\in\mathbb{R}$ such that
$\mu=\beta\lambda$ a.e. in $\Omega$ and $\theta=\lambda$ a.e. on $\Gamma$.
Integrating (3.4c) over $\Omega$ and (3.4d) over $\Gamma$, we infer that

$\lambda=\dfrac{\int_{\Omega}F^{\prime}\left(u\right)\,\mathrm{d}x+\int_{\Gamma}G^{\prime}\left(v\right)\,\mathrm{d}\Gamma}{\beta\left|\Omega\right|+\left|\Gamma\right|}\text{.}$
(4.1)

Therefore, we obtain that $\mathcal{N}_{m}^{L}$ consists of solutions to the
following stationary problem.

$\displaystyle-\Delta u+F^{\prime}\left(u\right)$ $\displaystyle=\beta\lambda$
$\displaystyle\quad\text{ in }\Omega,$ (4.2a)
$\displaystyle-\Delta_{\Gamma}v+G^{\prime}\left(v\right)+\partial_{n}u$
$\displaystyle=\lambda$ $\displaystyle\quad\text{ on }\Gamma,$ (4.2b)
$\displaystyle u|_{\Gamma}$ $\displaystyle=v$ $\displaystyle\quad\text{ on
}\Gamma,$ (4.2c)
$\displaystyle\textstyle\beta\int_{\Omega}u\,\mathrm{d}x+\int_{\Gamma}v\,\mathrm{d}\Gamma$
$\displaystyle=m.$ (4.2d)

Hence, in order to prove that $\mathcal{N}_{m}^{L}$ is nonempty, it suffices
to show that (4.2) has at least one solution. Since the problem (4.2) does not
depend on $L$, we notice that the set of stationary points
$\mathcal{N}_{m}^{L}$ is actually independent of $L$. Thus, in the remainder
of this section, we will just use the notation
$\mathcal{N}_{m}:=\mathcal{N}_{m}^{L}$.

In the following, we interpret the total free energy as a continuous
functional

$\displaystyle E:{\mathcal{V}}^{1}\to[0,\infty),\quad(u,v)\mapsto E(u,v)$
(4.3)

where for any $(u,v)\in{\mathcal{W}_{\beta,m}^{1}}$, the expression $E(u,v)$
is as defined in (3.2).

We first prove that the energy functional $E$ has a global minimizer on the
space ${\mathcal{W}_{\beta,m}^{1}}$. Next, we show that a minimizer of $E$ on
${\mathcal{W}_{\beta,m}^{1}}$ is actually a strong solution of the system
$(\eqref{5.33},\eqref{5.34})$ meaning that it is a stationary point.

######  Lemma 4.1.

Suppose that the assumptions ($\mathrm{A1}$)-($\mathrm{A3}$) hold. Then the
functional $E|_{{\mathcal{W}_{\beta,m}^{1}}}$ has at least one minimizer
$(u_{*},v_{*})\in{\mathcal{W}_{\beta,m}^{1}}$, i.e.,

$E(u_{*},v_{*})=\inf_{(u,v)\in{\mathcal{W}_{\beta,m}^{1}}}E\left(u,v\right).$

  * Proof.

To prove the assertion, we employ the direct method of calculus of variations.
From the assumptions ($\mathrm{A3}$) on $F$ and $G$, we obtain the lower bound

$E\left(u,v\right)\geq\frac{1}{2}\left\|\nabla
u\right\|_{L^{2}\left(\Omega\right)}^{2}+\frac{1}{2}\left\|\nabla_{\Gamma}v\right\|_{L^{2}\left(\Gamma\right)}^{2}+a_{F}\left\|u\right\|_{L_{p}\left(\Omega\right)}^{p}+a_{G}\left\|v\right\|_{L_{q}\left(\Gamma\right)}^{q}-b_{F}-b_{G}$
(4.4)

for all $(u,v)\in{\mathcal{W}_{\beta,m}^{1}}$. Hence, the infimum

${E}_{*}:=\underset{(u,v)\in{\mathcal{W}_{\beta,m}^{1}}}{\inf}E\left(u,v\right)$

exists and thus, we can find a minimizing sequence
$\left\\{(u_{k},v_{k})\right\\}_{k\in\mathbb{N}}\subset$
${\mathcal{W}_{\beta,m}^{1}}$ such that

$\lim_{k\rightarrow\infty}E\left(u_{k},v_{k}\right)={E}_{*}\qquad\text{and}\qquad
E\left(u_{k},v_{k}\right)\leq{E}_{*}+1\quad\text{for all $k\in\mathbb{N}$.}$
(4.5)

Combining (4.4) and (4.5), it follows that
$\left\\{\left(u_{k},v_{k}\right)\right\\}_{k\in\mathbb{N}}$ is bounded in
${\cal H}^{1}$. Then, according to the Banach-Alaoglu theorem, it holds that

$(u_{k},v_{k})\rightharpoonup({u}^{*},{u}^{*})\quad\text{in
$\mathcal{H}^{1}$}$ (4.6)

along a non-relabeled subsequence. Invoking Sobolev’s embedding theorem we
further deduce that

$\displaystyle\left(u_{k},v_{k}\right)\rightarrow\left({u}^{*},{v}^{*}\right)\quad\text{in
}L^{p}\left(\Omega\right)\times L^{q}\left(\Gamma\right),\text{ and a.e.~{}in
}\Omega\times\Gamma$ (4.7)

after another subsequence extraction, where $p$ and $q$ are the exponents
introduced in ($\mathrm{A3}$). Since $F$ and $G$ are continuous, (4.7) implies

$F\left(u_{k}\right)\rightarrow F\left({u}^{*}\right)\quad\text{a.e.~{}in
}\Omega,\quad\text{and}\quad G\left(v_{k}\right)\rightarrow
G\left({v}^{*}\right)\quad\text{a.e.~{}in }\Gamma.$

Recalling assumption ($\mathrm{A3}$), it follows from Lebesgue’s generalized
convergence theorem [2, Sect. 3.25] that

$\lim_{k\rightarrow\infty}\int_{\Omega}F\left(u_{k}\right)\,\mathrm{d}x=\int_{\Omega}F\left({u}^{*}\right)\,\mathrm{d}x,\quad\text{and}\quad\lim_{k\rightarrow\infty}\int_{\Gamma}G\left(v_{k}\right)\,\mathrm{d}\Gamma=\int_{\Gamma}G\left({v}^{*}\right)\,\mathrm{d}\Gamma.$

As all the other terms in the energy functional are convex, they are lower
semicontinuous, and we thus conclude that

$E\left(u^{*},v^{*}\right)\leq\underset{m\rightarrow\infty}{\lim\inf}\;E\left(u_{k},v_{k}\right)={E}_{*}.$

This proves that $(u^{*},v^{*})$ is a minimizer of the functional $E$ on
${\mathcal{W}_{\beta,m}^{1}}$. ∎

We further show that the energy functional is of class $C^{2}$.

######  Lemma 4.2 (Regularity of the energy functional).

The energy functional $E$ is twice continuously Fréchet differentiable. For
every $(u,v),(\zeta,\eta)\in{\mathcal{V}}^{1}$, we have

$\displaystyle\left\langle
E^{\prime}(u,v),(\zeta,\eta)\right\rangle_{{\mathcal{V}}^{1}}$
$\displaystyle=\int_{\Omega}\nabla
u\cdot\nabla\zeta+F^{\prime}(u)\zeta\,\mathrm{d}x+\int_{\Gamma}\nabla_{\Gamma}v\cdot\nabla_{\Gamma}\eta+G^{\prime}(v)\eta\,\mathrm{d}\Gamma.$
(4.8)

For every
$(u,v),(\zeta_{1},\eta_{1}),(\zeta_{2},\eta_{2})\in{\mathcal{V}}^{1}$, we have

$\displaystyle\begin{aligned} \left\langle
E^{\prime\prime}(u,v)(\zeta_{1},\eta_{1}),(\zeta_{2},\eta_{2})\right\rangle_{{\mathcal{V}}^{1}}&=\int_{\Omega}\nabla\zeta_{1}\cdot\nabla\zeta_{2}+F^{\prime\prime}(u)\zeta_{1}\zeta_{2}\,\mathrm{d}x\\\
&\quad+\int_{\Gamma}\nabla_{\Gamma}\eta_{1}\cdot\nabla_{\Gamma}\eta_{2}+G^{\prime\prime}(v)\eta_{1}\eta_{2}\,\mathrm{d}\Gamma.\end{aligned}$
(4.9)

The proof of this lemma is straightforward and will thus not be presented. For
a very similar result including a detailed proof, we refer to [13, Lem. 6.2].

######  Lemma 4.3 (Critical points of $E$ are steady states).

Suppose that the assumptions ($\mathrm{A1}$)-($\mathrm{A3}$) are satisfied and
suppose that $(u_{*},v_{*})\in{\mathcal{W}_{\beta,m}^{1}}$. Then, the
following statements are equivalent:

  1. $(\mathrm{i})$

$(u_{*},v_{*})$ is a critical point of $E|_{{\mathcal{W}_{\beta,m}^{1}}}$.

  2. $(\mathrm{ii})$

$(u_{*},v_{*})\in{\mathcal{W}_{\beta,m}^{2}}$ and $(u_{*},v_{*})$ is a strong
solution of the problem $(\eqref{5.33},\eqref{5.34})$. In particular, this
means that $(u_{*},v_{*})\in\mathcal{N}_{m}$.

  * Proof.

We first notice that (i) is equivalent to

$\displaystyle\begin{aligned} &\left\langle
E^{\prime}(u_{*},v_{*}),(\zeta,\xi)\right\rangle_{{\mathcal{V}}^{1}}\\\
&\quad=\int_{\Omega}\left(\nabla
u_{*}\cdot\nabla\zeta+F^{\prime}\left(u_{*}\right)\zeta\right)\,\mathrm{d}x+\int_{\Gamma}\left(\nabla_{\Gamma}v_{*}\cdot\nabla_{\Gamma}\xi+G^{\prime}\left(v_{*}\right)\xi\right)\,\mathrm{d}\Gamma=0\end{aligned}$
(4.10)

for all $(\zeta,\xi)\in{\mathcal{W}_{\beta,0}^{1}}$. Now, for any
$(\overline{\zeta},\overline{\xi})\in{\mathcal{V}}^{1}$, we define

$\gamma=\dfrac{\beta\int_{\Omega}\overline{\zeta}\,\mathrm{d}x+\int_{\Gamma}\overline{\xi}\,\mathrm{d}\Gamma}{\beta\left|\Omega\right|+\left|\Gamma\right|}.$

Hence,
$(\zeta,\xi)=(\overline{\zeta}-\gamma,\overline{\xi}-\gamma)\in{\mathcal{W}_{\beta,0}^{1}}\,$,
and thus (4.10) is equivalent to

$\displaystyle\begin{aligned} &\int_{\Omega}\left(\nabla
u_{*}\cdot\nabla\overline{\zeta}+F^{\prime}\left(u_{*}\right)\overline{\zeta}\right)\,\mathrm{d}x+\int_{\Gamma}\left(\nabla_{\Gamma}v_{*}\cdot\nabla_{\Gamma}\overline{\xi}+G^{\prime}\left(v_{*}\right)\overline{\xi}\right)\,\mathrm{d}\Gamma\\\
&\qquad=\int_{\Omega}F^{\prime}\left(u_{*}\right)\gamma\,\mathrm{d}x+\int_{\Gamma}G^{\prime}\left(v_{*}\right)\gamma\,\mathrm{d}\Gamma=\int_{\Omega}\beta\lambda\,\overline{\zeta}\,\mathrm{d}x+\int_{\Gamma}\lambda\,\overline{\xi}\,\mathrm{d}\Gamma\end{aligned}$
(4.11)

for all $(\overline{\zeta},\overline{\xi})\in{\mathcal{V}}^{1}$. This means
that $(u_{*},v_{*})\in{\mathcal{W}_{\beta,m}^{1}}$ is a weak solution of the
problem ((4.1),(4.2)). Moreover, recalling the growth assumption
($\mathrm{A3}$), we can use elliptic regularity theory for problems with bulk-
surface coupling (see [43, Thm. 3.3]) to conclude that
$(u_{*},v_{*})\in\mathcal{H}^{2}$. Therefore, $(u_{*},v_{*})$ is actually a
strong solution of the problem ((4.1),(4.2)) which directly yields
$(u_{*},v_{*})\in\mathcal{N}_{m}$. We conclude that (i) and (ii) are
equivalent and thus, the proof is complete. ∎

######  Lemma 4.4.

Suppose that the assumptions ($\mathrm{A1}$)-($\mathrm{A3}$) hold. Then, the
set of stationary points $\mathcal{N}_{m}$ is a nonempty and bounded subset of
${\mathcal{W}_{\beta,m}^{2}}$.

  * Proof.

Let $C$ denote a generic constant depending only on $\Omega$, $F$, $G$,
$\beta$ and $m$ which may change its value from line to line. We first notice
that Lemma 4.1 and Lemma 4.3 directly imply that $\mathcal{N}_{m}$ is
nonempty. Let now $(u,v)\in\mathcal{N}_{m}$ be arbitrary and let
$\lambda\in\mathbb{R}$ as defined in (4.1). Testing (4.2a) with $u$ and
$\eqref{5.34:2}$ with $v$, and summing the obtained identities we get

$\displaystyle\begin{aligned} &\left\|\nabla
u\right\|_{L^{2}\left(\Omega\right)}^{2}+\left\|\nabla_{\Gamma}v\right\|_{L^{2}\left(\Gamma\right)}^{2}+\int_{\Omega}F_{1}^{\prime}\left(u\right)u\,\mathrm{d}x+\int_{\Gamma}G_{1}^{\prime}\left(v\right)v\,\mathrm{d}\Gamma\\\
&\qquad=m\lambda-\int_{\Omega}F_{2}^{\prime}\left(u\right)u\,\mathrm{d}x-\int_{\Gamma}G_{2}^{\prime}\left(v\right)v\,\mathrm{d}\Gamma.\end{aligned}$
(4.12)

Performing a Taylor expansion, recalling that $F_{1}^{\prime\prime}$ and
$G_{1}^{\prime\prime}$ are non-negative, and invoking the growth conditions
(2.4a) and (2.4b) in ($\mathrm{A3}$), we deduce that

$\displaystyle F_{1}^{\prime}(s)\,s$ $\displaystyle\geq F_{1}(s)-F_{1}(0)$
$\displaystyle\geq a_{F^{\prime}}\left|s\right|^{p}-C,$ (4.13) $\displaystyle
G_{1}^{\prime}(s)\,s$ $\displaystyle\geq G_{1}(s)-G_{1}(0)$ $\displaystyle\geq
a_{G^{\prime}}\left|s\right|^{q}-C$ (4.14)

for all $s\in\mathbb{R}$. Moreover, recalling (4.1) and the growth assumptions
in ($\mathrm{A3}$), by means of Young’s inequality we infer that for any
$\alpha>0$ there exists a constant $C_{\alpha}$ depending only on $\Omega$,
$F$, $G$, $\beta$, $m$ and $\alpha$ such that

$\displaystyle\left|m\lambda\right|+\int_{\Omega}\left|F_{2}^{\prime}(u)\right|\left|u\right|\,\mathrm{d}x+\int_{\Gamma}\left|G_{2}^{\prime}(v)\right|\left|v\right|\,\mathrm{d}\Gamma\leq
C_{\alpha}+\alpha\big{(}\left\|u\right\|_{L^{p}\left(\Omega\right)}^{p}+\left\|v\right\|_{L^{q}\left(\Gamma\right)}^{q}\big{)}.$
(4.15)

Fixing $\alpha=\frac{1}{2}\min\\{a_{F^{\prime}},a_{G^{\prime}}\\}$ and using
the estimates (4.13)–(4.15), we conclude from (4.12) that

$\displaystyle\begin{aligned} &\left\|\nabla
u\right\|_{L^{2}\left(\Omega\right)}^{2}+\left\|\nabla_{\Gamma}v\right\|_{L^{2}\left(\Gamma\right)}^{2}+\frac{a_{F^{\prime}}}{2}\|u\|_{L^{p}(\Omega)}^{p}+\frac{a_{G^{\prime}}}{2}\|v\|_{L^{q}(\Gamma)}^{q}\leq
C.\end{aligned}$ (4.16)

This eventually yields $\left\|(u,v)\right\|_{\mathcal{H}^{1}}\leq C$. From
the growth conditions in ($\mathrm{A3}$) we infer
$\|F^{\prime}(u)\|_{L^{2}(\Omega)}\leq C$ and
$\|G^{\prime}(v)\|_{L^{2}(\Gamma)}\leq C$. Hence, we can apply regularity
theory for elliptic problems with bulk-surface coupling (see [43, Thm. 3.3])
to conclude that $\|(u,v)\|_{\mathcal{H}^{2}}\leq C$, which completes the
proof. ∎

####  4.1.2 Existence of a global attractor

We first prove the following asymptotic compactness property by exploiting the
smoothing estimate (3.7).

######  Lemma 4.5 (Asymptotic compactness).

Suppose that the assumptions ($\mathrm{A1}$)-($\mathrm{A3}$) are satisfied,
and let $B$ be a bounded subset of ${\mathcal{W}_{\beta,m}^{1}}$. Then for any
sequences $\left\\{(u_{k},v_{k})\right\\}_{k\in\mathbb{N}}\subset B$ and
$\\{t_{k}\\}_{k\in\mathbb{N}}\subset\mathbb{R}_{0}^{+}$ with $t_{k}\to\infty$
as $k\to\infty$, the sequence
$\left\\{S_{m}^{L}\left(t_{k}\right)(u_{k},v_{k})\right\\}_{k\in\mathbb{N}}$
has a strongly convergent subsequence in ${\mathcal{W}_{\beta,m}^{2}}$.

  * Proof.

We first obtain from the smoothing property (3.7) that for all
$k\in\mathbb{N}$ with $t_{k}\geq 1$,

$\displaystyle\left\|S_{m}^{L}\left(t_{k}\right)(u_{k},v_{k})\right\|_{\mathcal{H}^{3}}\leq
C_{*}\left(\dfrac{t_{k}+1}{t_{k}}\right)^{1/2}\leq\sqrt{2}\,C_{*}.$

Recall that the constant $C_{*}$ from (3.7) does not depend on $t_{k}$. Hence,
there exists $(u_{*},v_{*})\in\mathcal{H}^{3}$ such that
$S_{m}^{L}\left(t_{k}\right)(u_{k},v_{k})$ converges to $(u_{*},v_{*})$ weakly
in $\mathcal{H}^{3}$ as $k\to\infty$, up to extraction of a subsequence. As
$\mathcal{H}^{3}$ is compactly embedded in $\mathcal{H}^{2}$, we further
deduce that $S_{m}^{L}\left(t_{k}\right)(u_{k},v_{k})$ converges to
$(u_{*},v_{*})$ strongly in $\mathcal{H}^{2}$. Moreover, since
$S_{m}^{L}\left(t_{k}\right)(u_{k},v_{k})\in{\mathcal{W}_{\beta,m}^{2}}$ for
all $k\in\mathbb{N}$, we eventually conclude that
$(u_{*},v_{*})\in{\mathcal{W}_{\beta,m}^{2}}$ which proves the claim. ∎

######  Lemma 4.6 (Gradient system).

Suppose that ($\mathrm{A1}$)-($\mathrm{A3}$) hold. Then the energy functional
$E$ that was defined in (3.2) is a strict Lyapunov function for the dynamical
system $({\mathcal{W}_{\beta,m}^{1}},\\{S_{m}^{L}\left(t\right)\\}_{t\geq
0})$. This means that the dynamical system is a so-called _gradient system_.

  * Proof.

Let $(u^{L},v^{L},\mu^{L},\theta^{L})$ denote an arbitrary weak solution of
the system (3.1) on $\mathbb{R}_{0}^{+}$ to the initial datum
$(u_{0},v_{0})\in{\mathcal{W}_{\beta,m}^{1}}$ in the sense of Theorem 3.1. In
particular, this means that
$\big{(}u^{L}(t),v^{L}(t)\big{)}=S_{m}^{L}(t)(u_{0},v_{0})$ for all $t\geq 0$.
Interpreting $(u^{L},v^{L},\mu^{L},\theta^{L})$ as a weak solution starting at
any time $s\geq 0$ (instead of zero), the energy inequality (3.6) yields that
for almost all $t\in\mathbb{R}_{0}^{+}$ with $t\geq s$,

$\displaystyle\begin{aligned} &E\big{(}u^{L}(t),v^{L}(t)\big{)}\\\
&\;\;+\frac{1}{2}\int_{s}^{t}\big{\|}\nabla\mu^{L}(\tau)\big{\|}_{L^{2}(\Omega)}^{2}+\big{\|}\nabla_{\Gamma}\theta^{L}(\tau)\big{\|}_{L^{2}(\Gamma)}^{2}+\frac{1}{L}\big{\|}\beta\theta^{L}(\tau)-\mu^{L}(\tau)\big{\|}_{L^{2}(\Gamma)}^{2}\,\mathrm{d}\tau\\\
&\leq E\left(u^{L}(s),v^{L}(s)\right).\end{aligned}$ (4.17)

This implies that $E$ is nonincreasing along weak solutions of the system
(3.1). Hence, the energy functional $E$ is a Lyapunov function for the
dynamical system $\\{S_{m}^{L}\left(t\right)\\}_{t\geq 0}$.

To prove that the Lyapunov function $E$ is actually strict, we assume that
there exists a weak solution $(u^{L},v^{L},\mu^{L},\theta^{L})$ as well as
$s,t\in\mathbb{R}_{0}^{+}$ with $s<t$ such that
$(u^{L},v^{L})\notin\mathcal{N}_{m}$ and

$E\left(u^{L}(t),v^{L}(t)\right)=E\left(u^{L}(s),v^{L}(s)\right).$

Since, $E$ is nonincreasing along weak solutions, this directly implies that
$E(u^{L}(\cdot),v^{L}(\cdot))$ is constant on the time interval $[s,t]$. Thus,
the inequality (4.17) entails that

$\displaystyle\nabla\mu^{L}$ $\displaystyle=0$
$\displaystyle\quad\text{a.e.~{}in }\Omega\times[s,t],$ (4.18a)
$\displaystyle\nabla_{\Gamma}\theta^{L}$ $\displaystyle=0$
$\displaystyle\quad\text{a.e.~{}on }\Gamma\times[s,t],$ (4.18b)
$\displaystyle\beta\theta^{L}$ $\displaystyle=\mu^{L}$
$\displaystyle\quad\text{a.e.~{}on }\Gamma\times[s,t].$ (4.18c)

Plugging these relations into the weak formulations (3.4a) and (3.4b), we
infer that $u^{L}$ and $v^{L}$ are constant in time on $[s,t]$. It immediately
follows that $\mu^{L}$ and $\theta^{L}$ are also constant on $[s,t]$, and from
the uniqueness of weak solutions, we eventually conclude that
$(u^{L},v^{L},\mu^{L},\theta^{L})$ is constant on the whole time interval
$\mathbb{R}_{0}^{+}$. This means that $(u^{L},v^{L})$ solves (4.2) and thus,
$(u^{L},v^{L})\in\mathcal{N}_{m}$, which contradicts the assumptions.

We have thus proven that the energy functional $E$ is strictly decreasing
along non-stationary weak solutions of the system (3.1) and thus, $E$ is a
strict Lyapunov function for the dynamical system
$({\mathcal{W}_{\beta,m}^{1}},\\{S_{m}^{L}\left(t\right)\\}_{t\geq 0})$. ∎

To prove the existence of a global attractor which is bounded in
$\mathcal{W}_{\beta,m}^{3}$, we first recall the following definitions.

######  Definition 4.7 (Global attractor, cf. [15, Def. 7.2.1.]).

Let $\left(X,d\right)$ be a metric space and let $\left\\{S(t)\right\\}_{t\geq
0}$ be a semigroup on $X$. A set $\mathcal{A}\subset X$ is called a global
attractor for the dynamical system $(X,\\{S(t)\\}_{t\geq 0})$, if

  1. $(\mathrm{i})$

$\mathcal{A}$ is a compact subset of $X$,

  2. $(\mathrm{ii})$

$\mathcal{A}$ is invariant, i.e. $S(t)\mathcal{A}=\mathcal{A}$ for all $t\geq
0$,

  3. $(\mathrm{iii})$

$\underset{t\rightarrow\infty}{\lim}\mathrm{dist}_{X}(S(t)B,\mathcal{A})=0$
for every bounded set $B\subset X$.

In particular, it follows directly from this definition that a global
attractor is always unique.

######  Definition 4.8 (Unstable manifold, cf. [15, Def. 7.5.1.]).

Let $\mathcal{N}$ be the set of stationary points of the dynamical system
$\left(X,\\{S(t)\\}_{t\geq 0}\right)$. We define the unstable manifold
$\mathcal{M}_{\mathrm{u}}\left(\mathcal{N}\right)$ emanating from the set
$\mathcal{N}$ as the set of all $y\in X$ such that there exists a full
trajectory $\gamma=\\{u(t):t\in\mathbb{R}\\}$ where $u$ satisfies the
properties

$u(0)=y\text{ and
}\lim_{t\rightarrow-\infty}\mathrm{dist}_{X}(u(t),\mathcal{N})=0.$

From the results established above, we now deduce the existence of a unique
global attractor.

######  Theorem 4.9 (Existence of a unique global attractor).

Suppose that the assumptions ($\mathrm{A1}$)-($\mathrm{A3}$) are satisfied.
Then, the semigroup $\left\\{S_{m}^{L}(t)\right\\}_{t\geq 0}$ generated by the
weak solutions of the problem (3.1) possesses a unique global attractor
${\mathcal{A}}_{m}^{L}\subset{\mathcal{W}_{\beta,m}^{1}}$, and it holds that

$\displaystyle\mathcal{A}_{m}^{L}=\mathcal{M}^{L}_{\mathrm{u}}\left(\mathcal{N}_{m}\right).$
(4.19)

Moreover, the global attractor ${\mathcal{A}}_{m}^{L}$ is a bounded subset of
$\mathcal{W}_{\beta,m}^{3}$.

  * Proof.

Due to Lemma 4.4, Lemma 4.5 and Lemma 4.6, the existence of a unique global
attractor ${\mathcal{A}}_{m}^{L}\subset{\mathcal{W}_{\beta,m}^{1}}$ to the
dynamical system $({\mathcal{W}_{\beta,m}^{1}},S_{m}^{L}\left(t\right))$ which
can be expressed as
$\mathcal{A}_{m}^{L}=\mathcal{M}^{L}_{\mathrm{u}}\left(\mathcal{N}_{m}\right)$
follows directly from [15, Cor. 7.5.7]. The fact that $\mathcal{A}_{m}^{L}$ is
bounded in $\mathcal{H}^{3}$ is a direct consequence of the invariance
property in Definition 4.7(ii) and the smoothing property (3.7). ∎

####  4.1.3 Convergence to stationary points

The invariance property of the global attractor ensures that for any
$(u_{0},v_{0})\in\mathcal{A}_{m}^{L}$, the corresponding weak solution
$(u^{L},v^{L},\mu^{L},\theta^{L})$ exists on $\mathbb{R}$ and it holds that

$\displaystyle(u_{0},v_{0})\in\left\\{(u^{L}(t),v^{L}(t))\,\big{|}\,t\in\mathbb{R}\right\\}\subset\mathcal{A}_{m}^{L}.$

A detailed proof can be found, e.g., in [14, Lem. 6.1]. In particular, due to
Proposition 3.4(b), this weak solution satisfies $(u^{L},v^{L})\in
C(\mathbb{R};\mathcal{H}^{2})$. This allows the following definition:

######  Definition 4.10 ($\omega$-limit set and $\alpha$-limit set).

For any initial datum $(u_{0},v_{0})\in{\mathcal{W}_{\beta,m}^{1}}\,$, let
$(u^{L},v^{L},\mu^{L},\theta^{L})$ denote the corresponding weak solution of
the system (3.1). We define the set

$\displaystyle\omega^{L}\left(u_{0},v_{0}\right):=\left\\{(u_{*},v_{*})\in\mathcal{A}_{m}^{L}\;\middle|\;\begin{aligned}
&\text{$\exists\\{t_{k}\\}_{k\in\mathbb{N}}\subset\mathbb{R}_{0}^{+}$ with
$t_{k}\to\infty$ such that}\\\
&\text{$S^{L}_{m}\left(t_{k}\right)(u_{0},v_{0})\to(u_{*},v_{*})$ in
${\mathcal{W}_{\beta,m}^{2}}$}\\\ \end{aligned}\right\\},$

which is called the _$\omega$ -limit set_ of $(u_{0},v_{0})$. If
$(u_{0},v_{0})\in\mathcal{A}^{L}_{m}$, we further define

$\displaystyle\alpha^{L}\left(u_{0},v_{0}\right):=\left\\{(u_{*},v_{*})\in\mathcal{A}_{m}^{L}\;\middle|\;\begin{aligned}
&\text{$\exists\\{t_{k}\\}_{k\in\mathbb{N}}\subset\mathbb{R}$ with
$t_{k}\to-\infty$ such that}\\\
&\text{$\big{(}u^{L}(t_{k}),v^{L}(t_{k})\big{)}\to(u_{*},v_{*})$ in
${\mathcal{W}_{\beta,m}^{2}}$}\\\ \end{aligned}\right\\},$

which is referred to as the _$\alpha$ -limit set_ of $(u_{0},v_{0})$.

For our purposes, the following properties will be essential.

######  Lemma 4.11 (Properties of the limit sets).

The limit sets have the following properties:

  1. $(\mathrm{a})$

For any $(u_{0},v_{0})\in{\mathcal{W}_{\beta,m}^{1}}\,$, the set
$\omega^{L}\left(u_{0},v_{0}\right)$ is nonempty, compact in
${\mathcal{W}_{\beta,m}^{2}}$, and invariant in the sense of Definition
4.7(ii). Moreover, it holds that

$\displaystyle\lim_{t\to\infty}\mathrm{dist}_{\mathcal{H}^{2}}\big{(}S^{L}_{m}(t)(u_{0},v_{0}),\omega^{L}(u_{0},v_{0})\big{)}=0,\quad\omega^{L}(u_{0},v_{0})\subset\mathcal{N}_{m}\subset{\mathcal{W}_{\beta,m}^{2}}.$
(4.20)

  2. $(\mathrm{b})$

For any $(u_{0},v_{0})\in\mathcal{A}^{L}_{m}\,$, the set
$\alpha^{L}\left(u_{0},v_{0}\right)$ is nonempty, compact in
${\mathcal{W}_{\beta,m}^{2}}$, and invariant in the sense of Definition
4.7(ii). Moreover, it holds that

$\displaystyle\lim_{t\to-\infty}\mathrm{dist}_{\mathcal{H}^{2}}\big{(}S^{L}_{m}(t)(u_{0},v_{0}),\alpha^{L}(u_{0},v_{0})\big{)}=0,\quad\alpha^{L}(u_{0},v_{0})\subset\mathcal{N}_{m}\subset{\mathcal{W}_{\beta,m}^{2}}.$
(4.21)

  * Proof.

The assertions in (a) follow directly from [38, Thm. 4.3.3] and [10, Thm.
9.2.7] thanks to the asymptotic compactness result established in Lemma 4.5.

It is well-known that the $\alpha$-limit set can be expressed as

$\displaystyle\alpha^{L}(u_{0},v_{0})=\bigcap_{n\in\mathbb{N}}\overline{\phantom{\Big{|}}\big{\\{}\big{(}u^{L}(t),v^{L}(t)\big{)}\;\big{|}\;t\leq-n\big{\\}}}^{\mathcal{H}^{2}},$

see, e.g., [15, Eq. (7.1.4)]. As the intersection of closed sets is also
closed, we infer that $\alpha^{L}(u_{0},v_{0})$ is a closed subset of
$\mathcal{A}^{L}_{m}$. Hence, since $\mathcal{A}^{L}_{m}$ is compact in
${\mathcal{W}_{\beta,m}^{2}}$ so is $\alpha^{L}(u_{0},v_{0})$. Recalling
Definition 4.8, the assertion (4.21) follows directly from Theorem 4.9 since
$\mathcal{A}_{m}^{L}=\mathcal{M}^{L}_{\mathrm{u}}\left(\mathcal{N}_{m}\right)$.
This proves (b) and thus, the proof is complete. ∎

We will now show that for every
$(u_{0},v_{0})\in{\mathcal{W}_{\beta,m}^{1}}\,$, the corresponding state
$(u^{L}(t),v^{L}(t))=S_{m}^{L}(t)(u_{0},v_{0})$ converges to one single
stationary point $(u_{\infty},v_{\infty})\in\mathcal{N}_{m}$ as
$t\rightarrow\infty$. In addtion, we will prove that for every
$(u_{0},v_{0})\in\mathcal{A}_{m}^{L}\,$, the pair $(u^{L}(t),v^{L}(t))$
converges to a single stationary point
$(u_{-\infty},v_{-\infty})\in\mathcal{N}_{m}$ as $t\rightarrow-\infty$.

Arguing as in [13, Prop. 6.6] we next establish a Łojasiewicz–Simon type
inequality. Here, we restrict ourselves to dealing only with the case $m=0$.
To see that this is not really a restriction, we introduce the constant
$M:=m/(\beta\left|\Omega\right|+\left|\Gamma\right|)$. If now
$(u^{L},v^{L},\mu^{L},\theta^{L})$ is a weak solution of the system (3.1) with
$(u^{L}(t),v^{L}(t))\in{\mathcal{W}_{\beta,m}^{1}}$ for all $t\geq 0$, then
the shifted quadruplet

$(u^{L}_{M},v^{L}_{M},\mu^{L}_{M},\theta^{L}_{M}):=(u^{L}-M,v^{L}-M,\mu^{L},\theta^{L})$

is in fact a weak solution of the problem (3.1) written for the modified
potentials $F_{M}:=F(\,\cdot\,+M)$ and $G_{M}:=G(\,\cdot\,+M)$ instead of $F$
and $G$. By construction, this solution satisfies
$(u_{M}^{L}(t),v_{M}^{L}(t))\in\mathcal{W}_{\beta,0}^{1}$ for all $t\geq 0$.
Therefore, the choice $m=0$ does not mean any loss of generality.

######  Lemma 4.12 (Łojasiewicz–Simon inequality).

Suppose that ($\mathrm{A4}$) holds, and that
$(u_{*},v_{*})\in{\mathcal{W}_{\beta,0}^{2}}$ is a critical point of the
functional $E|_{{\mathcal{W}_{\beta,0}^{1}}}$. Then, there exist constants
$\gamma\in\left(0,\frac{1}{2}\right]$ and $C,\sigma>0$ depending only on $L$,
$\beta$, $\Omega$ and $(u_{*},v_{*})$ such that for all
$(u,v)\in{\mathcal{W}_{\beta,0}^{2}}$ with
$\|(u,v)-(u_{*},v_{*})\|_{{\mathcal{W}_{\beta,0}^{1}}}<\sigma$, the
Łojasiewicz–Simon inequality

$\displaystyle\left|E(u,v)-E(u_{*},v_{*})\right|^{1-\gamma}\leq
C\|E^{\prime}(u,v)\|_{({\mathcal{W}_{\beta,0}^{1}})^{\prime}}$ (4.22)

is satisfied.

Comment. According to Lemma 4.3, any critical point
$(u_{*},v_{*})\in{\mathcal{W}_{\beta,0}^{1}}$ of the energy $E$ is actually a
stationary point, i.e., the regularity
$(u_{*},v_{*})\in\mathcal{N}_{0}\subset{\mathcal{W}_{\beta,0}^{2}}$ holds
automatically.

  * Proof.

To prove the lemma, we will show that the abstract result in [12, Cor. 3.11]
can be applied. In the following, we consider the restriction of the energy
functional to the linear subspace ${\mathcal{W}_{\beta,0}^{1}}$ of
${\mathcal{V}}^{1}$, and we write $\bar{E}:=E|_{{\mathcal{W}_{\beta,0}^{1}}}$,
$\bar{E}^{\prime}:=E^{\prime}|_{{\mathcal{W}_{\beta,0}^{1}}}$ and
$\bar{E}^{\prime\prime}:=E^{\prime\prime}|_{{\mathcal{W}_{\beta,0}^{1}}}$.

We first notice that Sobolev’s embedding theorem yields
${\mathcal{W}_{\beta,0}^{2}}\subset\mathcal{L}^{\infty}$. Hence, the
restriction of $\bar{E}^{\prime}$ to ${\mathcal{W}_{\beta,0}^{2}}$ is analytic
with values in $\mathcal{L}^{2}$. For more a more detailed reasoning in a
similar situation we refer to [12, Cor. 4.6]. For arbitrary
$(u,v)\in{\mathcal{W}_{\beta,0}^{2}}$, we regard the second-order Fréchet
derivative $\bar{E}^{\prime\prime}$ as the linearization of $\bar{E}^{\prime}$
which means

$\displaystyle\bar{E}^{\prime\prime}(u,v):{\mathcal{W}_{\beta,0}^{1}}\to({\mathcal{W}_{\beta,0}^{1}})^{\prime},\quad(\zeta_{1},\eta_{1})\mapsto\bar{E}^{\prime\prime}(u,v)(\zeta_{1},\eta_{1}).$

In view of (4.9), $\bar{E}^{\prime\prime}(u,v)$ can be interpreted as a
continuous, symmetric, elliptic differential operator associated with the
bilinear form

$\displaystyle\left\langle\bar{E}^{\prime\prime}(u,v)(\zeta_{1},\eta_{1}),(\zeta_{2},\eta_{2})\right\rangle_{{\mathcal{W}_{\beta,0}^{1}}}$
$\displaystyle\quad=\int_{\Omega}\nabla\zeta_{1}\cdot\nabla\zeta_{2}+F^{\prime\prime}(u)\zeta_{1}\zeta_{2}\,\mathrm{d}x+\int_{\Gamma}\nabla_{\Gamma}\eta_{1}\cdot\nabla_{\Gamma}\eta_{2}+G^{\prime\prime}(v)\eta_{1}\eta_{2}\,\mathrm{d}\Gamma,\quad(\zeta_{1},\eta_{1}),(\zeta_{2},\eta_{2})\in{\mathcal{W}_{\beta,0}^{1}},$

which results from (4.9). We further interpret the restriction of
$\bar{E}^{\prime\prime}(u,v)$ to ${\mathcal{W}_{\beta,0}^{2}}$ as

$\displaystyle\bar{E}^{\prime\prime}(u,v)|_{{\mathcal{W}_{\beta,0}^{2}}}:{\mathcal{W}_{\beta,0}^{2}}\to\mathcal{L}^{2},\quad(\zeta_{1},\eta_{1})\mapsto\bar{E}^{\prime\prime}(u,v)(\zeta_{1},\eta_{1}).$

Invoking the Lax–Milgram theorem, we deduce that the operator
$\bar{E}^{\prime\prime}(u,v)$ has a nontrivial resolvent set. In particular,
there exists at least one $\lambda>0$ such that the linear map

$\displaystyle\big{(}\bar{E}^{\prime\prime}(u,v)+\lambda\text{id}\big{)}^{-1}:({\mathcal{W}_{\beta,0}^{1}})^{\prime}\to({\mathcal{W}_{\beta,0}^{1}})^{\prime}$

is well-defined. As the embedding
${\mathcal{W}_{\beta,0}^{1}}\hookrightarrow({\mathcal{W}_{\beta,0}^{1}})^{\prime}$
is compact, we conclude that this linear operator is compact.

By regularity theory for elliptic problems with bulk-surface coupling (see
[43, Thm. 3.3]), we conclude that the kernel
$\text{Ker}\,\bar{E}^{\prime\prime}(u,v)$ belongs to
${\mathcal{W}_{\beta,0}^{2}}$. We further recall that the embedding
${\mathcal{W}_{\beta,0}^{2}}\hookrightarrow\mathcal{L}^{2}$ is compact.

Now, proceeding as in [24, Sect. 6.2, Thm. 4], the Fredholm alternative
implies that the kernel $\text{Ker}\,\bar{E}^{\prime\prime}(u,v)$ is finite
dimensional, and that the ranges $\text{Rg}\,\bar{E}^{\prime\prime}(u,v)$ and
$\text{Rg}\,\bar{E}^{\prime\prime}(u,v)\big{|}\raisebox{-1.0pt}{\scriptsize${\mathcal{W}_{\beta,0}^{2}}$}$
are closed in $({\mathcal{W}_{\beta,0}^{1}})^{\prime}$ and $\mathcal{L}^{2}$,
respectively. Furthermore, $({\mathcal{W}_{\beta,0}^{1}})^{\prime}$ can be
expressed as the orthogonal sum of $\text{Ker}\,\bar{E}^{\prime\prime}(u,v)$
and $\text{Rg}\,\bar{E}^{\prime\prime}(u,v)$, whereas $\mathcal{L}^{2}$ can be
expressed as the orthogonal sum of $\text{Ker}\,\bar{E}^{\prime\prime}(u,v)$
and
$\text{Rg}\,\bar{E}^{\prime\prime}(u,v)\big{|}\raisebox{-1.0pt}{\scriptsize${\mathcal{W}_{\beta,0}^{2}}$}$.
In particular, we can define a continuous orthogonal projection
$P:{\mathcal{W}_{\beta,0}^{1}}\rightarrow{\mathcal{W}_{\beta,0}^{1}}$ with
$\text{Rg}\,P=\text{Ker}\,\bar{E}^{\prime\prime}(u,v)$.

Given these considerations, we can conclude the proof by applying the results
in [12, Cor. 3.11] with $X={\mathcal{W}_{\beta,0}^{2}}$,
$V={\mathcal{W}_{\beta,0}^{1}}$, $Y=\mathcal{L}^{2}$, and
$W=({\mathcal{W}_{\beta,0}^{1}})^{\prime}$. ∎

######  Theorem 4.13 (Convergence to a single stationary point as
$t\to\infty$).

Suppose that the assumptions ($\mathrm{A1}$)–($\mathrm{A4}$) are satisfied.
Then, for any $(u_{0},v_{0})\in{\mathcal{W}_{\beta,m}^{1}}$ there exists a
unique stationary point $(u_{\infty},v_{\infty})\in\mathcal{N}_{m}$ such that

$\displaystyle\lim_{t\to\infty}\big{\|}S_{m}^{L}(t)(u_{0},v_{0})-(u_{\infty},v_{\infty})\big{\|}_{\mathcal{H}^{2}}=0.$
(4.23)

In particular, this means that
$\omega^{L}(u_{0},v_{0})=\\{(u_{\infty},v_{\infty})\\}\subset\mathcal{N}_{m}$.

  * Proof.

As discussed above, it suffices to handle the case $m=0$. Let
$(u_{0},v_{0})\in{\mathcal{W}_{\beta,0}^{1}}$ be arbitrary, and let
$(u^{L},v^{L},\mu^{L},\theta^{L})$ denote the corresponding weak solution of
the system (3.1). To prove the existence of a unique limit
$(u_{\infty},v_{\infty})\in\mathcal{N}_{0}$, it suffices to show that the
$\omega$-limit set of $(u_{0},v_{0})$ consists of one single point.