# A qualitative study of the generalized dispersive systems with time–delay:
The unbounded case

Roberto de A. Capistrano–Filho  Departamento de Matemática, Universidade
Federal de Pernambuco,
S/N Cidade Universitária, 50740-545, Recife (PE), Brazil
Email address<EMAIL_ADDRESS>,  Fernando Gallego
Departamento de Matemática y Estadística, Universidad Nacional de Colombia
(UNAL),
Cra 27 No. 64-60, 170003, Manizales, Colombia
Email address<EMAIL_ADDRESS>and  Vilmos Komornik  Département de
Mathématique, Université de Strasbourg,
7 rue René Descartes, 67084 Strasbourg Cedex, France
Email address<EMAIL_ADDRESS>
###### Abstract.

We study the asymptotic behavior of the solutions of the time-delayed higher-
order dispersive nonlinear differential equation

$u_{t}(x,t)+Au(x,t)+\lambda_{0}(x)u(x,t)+\lambda(x)u(x,t-\tau)=0$

where

$Au=(-1)^{j+1}\partial_{x}^{2j+1}u+(-1)^{m}\partial_{x}^{2m}u+\frac{1}{p+1}\partial_{x}u^{p+1}$

with $m\leq j$ and $1\leq p<2j$. Under suitable assumptions on the time delay
coefficients, we prove that the system is exponentially stable if the
coefficient of the delay term is bounded from below by a suitable positive
constant, without no assumption on the sign of the coefficient of the
undelayed feedback. Additionally, in the absence of delay, general results of
stabilization are established in $H^{s}(\mathbb{R})$ for $s\in[0,2j+1]$. Our
results generalize several previous theorems in the literature for the
Korteweg-de Vries type delayed systems.

###### Key words and phrases:

KdV-type equation, Damping mechanism, Delayed feedback, Stabilization,
Lyapunov approach

###### 2010 Mathematics Subject Classification:

35Q53, 93D15,93D30,93C20

*Corresponding author.

Capistrano–Filho was supported by CNPq grants 307808/2021-1, 401003/2022-1 and
200386/2022-0, CAPES-PRINT grant 88881.311964/2018-01, CAPES-MATHAMSUD grant
88881.520205/2020-01. and MATHAMSUD 21-MATH-03. Gallego was supported by
MATHAMSUD 21-MATH-03. Komornik was supported by the grant NSFC No. 11871348
and MATHAMSUD 21-MATH-03.

###### Contents

  1. 1 Introduction
    1. 1.1 Model description
    2. 1.2 Review of the results in the literature
    3. 1.3 Main results
    4. 1.4 Comments and structure of the article
  2. 2 Well-posedness
    1. 2.1 Linear system
    2. 2.2  Non-homogeneous system
    3. 2.3 Nonlinear estimates
    4. 2.4 Nonlinear system
      1. 2.4.1 Well-posedness in $L^{2}({{\mathbb{R}}})$
      2. 2.4.2 Well-posedness in $H^{2j+1}({\mathbb{R}})$
    5. 2.5 Interpolation arguments
  3. 3 Exponential stabilization: Damping and delayed system
    1. 3.1 Asymptotic behavior: Linearized system with localized damping
    2. 3.2 Asymptotic behavior: Nonlinear system
    3. 3.3 Asymptotic behavior: Indefinite damping
  4. 4 Exponential stabilization: Damping system
    1. 4.1 Stabilization in $H^{2j+1}({\mathbb{R}})$
    2. 4.2 Stabilization in $H^{s}({\mathbb{R}})$
  5. 5 Concluding remarks
    1. 5.1 Weak versus strong damping mechanism
    2. 5.2 General framework

##  1\. Introduction

###  1.1. Model description

Under suitable assumptions on amplitude, wavelength, wave steepness, and so
on, the study on asymptotic models for water waves has been extensively
investigated to understand the full water wave system, see, for instance, [1,
6, 7] and references therein for a rigorous justification of various
asymptotic models for surface and internal waves.

Formulating the waves as a free boundary problem of the incompressible,
irrotational Euler equation in an appropriate non-dimensional form, one has
two non-dimensional parameters $\delta:=\frac{h}{\lambda}$ and
$\varepsilon:=\frac{a}{h}$, where the water depth, the wavelength and the
amplitude of the free surface are parameterized as $h,\lambda$ and $a$,
respectively. Moreover, another non-dimensional parameter $\mu$ is called the
Bond number, which measures the importance of gravitational forces compared to
surface tension forces. The physical condition $\delta\ll 1$ characterizes the
waves, called long waves or shallow water waves. In particular, considering
the relations between $\varepsilon$ and $\delta$, we can have two well-known
regimes:

  1. (1)

Korteweg-de Vries (KdV): $\varepsilon=\delta^{2}\ll 1$ and
$\mu\neq\frac{1}{3}$. Under this regime, Korteweg and de Vries [24]111This
equation was indeed first introduced by Boussinesq [9], and Korteweg and de
Vries rediscovered it twenty years later. derived the following equation well-
known as a central equation among other dispersive or shallow water wave
models called the KdV equation from the equations for capillary-gravity waves:

$\pm 2u_{t}+3uu_{x}+\left(\frac{1}{3}-\mu\right)u_{xxx}=0.$

  2. (2)

Kawahara: $\varepsilon=\delta^{4}\ll 1$ and
$\mu=\frac{1}{3}+\nu\varepsilon^{\frac{1}{2}}$. In connection with the
critical Bond number $\mu=\frac{1}{3}$, Hasimoto [16] derived a fifth-order
KdV equation of the form

$\pm 2u_{t}+3uu_{x}-\nu u_{xxx}+\frac{1}{45}u_{xxxxx}=0,$

which is nowadays called the Kawahara equation.

In the last years, many authors have been interested to find the behavior of
solutions for the time-delayed KdV equation, time-delayed Kawahara equation,
and other time-delayed dispersive systems see, for instance, [2, 3, 11, 12,
21, 32] and the reference therein. In this article, we try to find general
results for a higher-order dispersive equation which in some sense recovers
the equations mentioned in the cited articles.

Due to this advance for this type of dispersive equation, our main focus is to
investigate the higher-order extension, for example, of KdV and Kawahara
equations, precisely to consider the Cauchy problem for the following higher-
order KdV-type equation posed in $\mathbb{R}$, namely

(1.1)
$\begin{cases}u_{t}(x,t)+(-1)^{j+1}\partial_{x}^{2j+1}u(x,t)+\frac{1}{2}\partial_{x}(u^{2})=0,&(t,x)\in\mathbb{R}\times\mathbb{R}.\\\
u(0,x)=u_{0}(x),&x\in\mathbb{R}.\end{cases}$

Specifically, (1.1) is called KdV and fifth-order KdV–type equation when $j=1$
and $j=2$, respectively. Or more generally, our goal is to prove the
stabilization results of the solutions for a time-delayed higher-order
dispersive system with a strong dissipative term, namely

(1.2)
$\begin{cases}u_{t}(x,t)+(-1)^{j+1}\partial_{x}^{2j+1}u(x,t)+(-1)^{m}\partial_{x}^{2m}u(x,t)+\lambda_{0}(x)u(x,t)\\\
\hskip
113.81102pt+\lambda(x)u(x,t-\tau)+{\color[rgb]{0,0,0}\frac{1}{p+1}\partial_{x}u^{p+1}(x,t)}=0&\quad\text{in}\quad\mathbb{R}\times(0,\infty),\\\
u(x,0)=u_{0}(x)&\quad\text{in}\quad\mathbb{R},\\\
u(x,s)=u_{0}(x,s)&\quad\text{in}\quad\mathbb{R}\times[-\tau,0].\end{cases}$

with $m\leq j$, $j,m\in\mathbb{N},$ and $1\leq p<2j$. Here, the constant
$\tau>0$ is the time delay and the coefficients are considered with the
following regularity

$\lambda_{0}(x),\lambda(x)\in L^{\infty}(\mathbb{R}).$

Thus, our main aim is to furnish sufficient conditions on the coefficients
$\lambda,\lambda_{0}$ to have well-posedness of the model (1.2) and
exponential decay estimates.

###  1.2. Review of the results in the literature

Let us briefly discuss the preceding works [2, 3, 11, 12, 21, 32] and the
results concerning the well-posedness of (1.1).

The local and global well-posedness of (1.1) was widely studied. The local
well-posedness result was first proved by Gorsky and Himonas [15] for
$s\geq-\frac{1}{2}$, and Hirayama [17] improved this for $s\geq-\frac{j}{2}$.
Both works were based on the standard Fourier restriction norm method.
Hirayama improved the bilinear estimate by using the factorization of the
resonant function.

The global well-posedness of (1.1) was established for $j=1,2$ by Colliander
et al. [13] and Kato [19], respectively, via ”I-method”. In [18] the authors
extended the results of [13] and [19] to $j\geq 3$. Their method follows the
argument in [13] for the periodic KdV equation, while some estimates are
slightly different. They proved that the IVP (1.1) is globally well-posed in
$H^{s}({\mathbb{T}})$ for $j\geq 3$ and $s\geq-\frac{j}{2}$.

To our knowledge, the only work concerning the control and stabilization
properties for the system (1.1) was done by the first author with two
collaborators in [10]. The authors studied the local and global control
results for (1.1) posed on the unit circle. More precisely, they considered
the system

$\begin{cases}\partial_{t}u+(-1)^{j+1}\partial_{x}^{2j+1}u+\frac{1}{2}\partial_{x}(u^{2})=f(t,x),&(t,x)\in{\mathbb{R}}\times{\mathbb{T}},\\\
u(0,x)=u_{0}(x),&x\in H^{s}({\mathbb{T}}),\end{cases}$

posed on a periodic domain $\mathbb{T}$. They showed the globally
controllability in $H^{s}({\mathbb{T}})$, for $s\geq 0$ by the forcing term

(1.3) $f(t,x):=g(x)\left(h(t,x)-\int_{\mathbb{T}}g(y)h(t,y)\;dy\right),$

supported in a given open set $\omega\subset\mathbb{T}$, where $g$ is a given
nonnegative smooth function satisfying

$\int_{\mathbb{T}}g\left(x\right)dx=1\quad\text{and}\quad\omega:=\left\\{g>0\right\\},$

and $h$ is the control input.

Considering the particular cases of the system (1.2), without delay and
damping terms, we recall some results from the literature concerning the
asymptotic properties. The well-known Korteweg–de Vries–Burgers equation

(1.4)
$u_{t}+u_{xxx}-u_{xx}+uu_{x}=0\quad\text{in}\quad\mathbb{R}\times(0,\infty),$

corresponding to the case $j=m=1$, was extensively studied. For example, in
[4] Amick et. al proved that the $L^{2}-$norm of solutions to (1.4) tends to
zero as $t\rightarrow\infty$ in a polynomial way, namely

$\left\lVert u(\cdot,t)\right\rVert_{2}\leq Ct^{-\frac{1}{2}}\quad\text{for
all}\quad t>0,$

with a positive constant $C.$ More recently, Cavalcanti et al. [12] studied a
damped KdV–Burgers equation,

(1.5)
$\begin{cases}u_{t}(x,t)+u_{xxx}(x,t)-u_{xx}(x,t)+\lambda_{0}(x)u(x,t)+u(x,t)u_{x}(x,t)=0&\quad\text{in}\quad\mathbb{R}\times(0,\infty),\\\
u(x,0)=u_{0}(x)&\quad\text{in}\quad\mathbb{R}.\end{cases}$

Under appropriate conditions on the damping coefficient $\lambda_{0}$, the
authors established its well-posedness and exponential stability for
indefinite damping $\lambda_{0}(x),$ giving exponential decay estimates on the
$L^{2}-$norm of solutions. In [14], the results of [12] were extended by
generalizing the nonlinear term of (1.5), and proving the global well-
posedness and exponential stabilization of the generalized KdV-Burger equation
under the presence of the localized and indefinite damping. In [21] the
authors consider the KdV-Burgers equation (1.5) in presence of a delayed
feedback $\lambda(x)u(x,t-\tau)$. They considered the system with damping and
delay feedback, showing the exponential decay estimates under appropriate
conditions on the damping coefficients.

It is important to point out the exponential decay estimates obtained in [26,
29] for the KdV equation posed in an interval with localized damping. Also,
periodic conditions have been considered in [20, 23] while more general
nonlinearities have been considered in [31]. Additionally, the robustness
concerning the delay of the boundary stability of the nonlinear KdV equation
has been studied in [5]. The authors obtain, under an appropriate condition on
the feedback, with and without delay, the local stabilization results for the
KdV equation with noncritical length. Moreover, in [32], the authors extend
this result for the nonlinear Korteweg-de Vries equation in the presence of an
internal delayed term.

In two recent articles [2, 3] the authors studied the qualitative and
numerical analysis of the nonlinear fourth-order delayed dispersive equation
in a bounded domain $I=[0,L]$ with boundary and initial conditions:

(1.6) $\begin{cases}u_{t}(x,t)-\sigma u_{xx}(x,t)+\mu
u_{xxxx}(x,t)+u(x,t-\tau)u_{x}(x,t)=0,&(x,t)\in I\times(0,\infty),\\\
u(0,t)=u(L,t)=u_{x}(0,t)=u_{x}(L,t)=0,&t>0,\\\ u(x,h)=f(x,h),&(x,h)\in
I\times[-\tau,0].\end{cases}$

The well-posedness, as well as the exponential stability of the zero solution
of (1.6), was established in [2]. The main ingredient of the proof was the
exploitation of the Schauder Fixed Point Theorem. This improved an earlier
result [3] in the sense that no interior damping control was required.
Additionally, numerical simulations were also presented in this article to
illustrate the theoretical result.

Finally, consider the Kawahara equation posed on a bounded interval under the
presence of localized damping and delay terms:

(1.7)
$\begin{cases}u_{t}(x,t)+u_{x}(x,t)+u_{xxx}(x,t)-u_{xxxxx}(x,t)+u(x,t)u_{x}(x,t)\\\
\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+a\left(x\right)u(x,t)+b(x)u(x,t-h)=0&x\in(0,L),~{}t>0,\\\
u\left(0,t\right)=u\left(L,t\right)=u_{x}\left(0,t\right)=u_{x}\left(L,t\right)=u_{xx}\left(L,t\right)=0&t>0,\\\
u\left(x,0\right)=u_{0}\left(x\right)&x\in(0,L),\\\
u(x,t)=z_{0}(x,t)&x\in(0,L),~{}t\in(-h,0).\end{cases}$

In a recent paper [11] the authors proved its exponential stabilization under
suitable assumptions. First, they showed the Kawahara system is exponentially
stable under some restriction of the spatial length of the domain. Next, they
introduced a more general delayed system and suitable energy functions, they
proved, via the Lyapunov approach, the exponential stability for small initial
data small, under a restriction on the spatial length of the domain. Then they
used a compactness-uniqueness argument to remove these hypotheses, thereby
obtaining a semi-global stabilization result.

We end our review with a recent paper [22] where abstract linear and nonlinear
evolutionary systems with feedback were studied. Specifically, the authors
considered the system

(1.8)
$\left\\{\begin{array}[]{ll}U^{\prime}(t)=AU(t)+k(t)BU(t-\tau)+F(U(t)),&\text{
in }(0,+\infty).\\\ U(0)=U_{0},&\text{ for }t\in(0,\tau),\\\
BU(t-\tau)=f(t),&\end{array}\right.$

where $A$ generates an exponentially stable semigroup in a Hilbert space $H$,
$B$ is a continuous linear operator of $H$ into itself, $k(t)\in
L^{1}_{loc}(0,+\infty)$ and $\tau>0$ is a delay parameter, with
$F:H\rightarrow H$ is a Lipschitz function. The main purpose of [22] was to
give a well-posedness result and an exponential decay estimate for the model
(1.8) with a damping coefficient $k(t)$ belonging only to $L^{1}_{loc}$. By
using the semigroup approach combined with Gronwall’s inequality, under some
mild assumptions on the involved functions and parameters, the well-posedness
of the problem was established and an exponential decay estimate was obtained.

###  1.3. Main results

With this state of the art, let us now present our main results which give a
necessary next step to understanding the asymptotic behavior for generalized
dispersive systems.

From now on, for sake of simplicity, the norms in the spaces
$L^{p}(\mathbb{R})$ and $L^{\infty}(\mathbb{R})$ will be denoted by
$\|\cdot\|_{p}$ and $\|\cdot\|_{\infty}$, respectively. Furthermore, we
introduce the Banach space

(1.9) ${\mathcal{B}}_{s,T}:=C([0,T];H^{s}(\mathbb{R}))\cap
L^{2}(0,T;H^{s+j}(\mathbb{R}))$

with the natural norm

$\|u\|_{\mathcal{B}_{s,T}}=\|u\|_{C([0,T];H^{s}(\mathbb{R}))}+\|\partial^{s+j}_{x}u\|_{L^{2}(0,T;L^{2}(\mathbb{R}))}.$

For $s=0$ we omit the subscript $s$, so that
$\mathcal{B}_{T}=\mathcal{B}_{0,T}$. Our first result ensures that (1.2) is
well-posed in the space $H^{s}({\mathbb{R}})$ for $0\leq s\leq 2j+1$ and
$1\leq p<2j$.

######  Theorem 1.1.

Let $T>0$. Consider $1\leq p<2j$, $j\geq 1$, and $0\leq s\leq 2j+1$ be given.
In addition, assume that $\lambda_{0},\lambda\in L^{\infty}({\mathbb{R}})$
when $s=0$ and $\lambda_{0},\lambda\in H^{j}({\mathbb{R}})$ when $s>0$. Then,
for any $u_{0}\in H^{s}({\mathbb{R}})$, the IVP (1.2) admits a unique solution
$u\in\mathcal{B}_{s,T}$. Moreover, there exists a nondecreasing continuous
function $\beta_{s}:{\mathbb{R}}^{+}\rightarrow{\mathbb{R}}^{+}$, such that

$\|u\|_{{\mathcal{B}}_{s,T}}\leq\beta_{s}(\|u_{0}\|_{2})\|u_{0}\|_{H^{s}({\mathbb{R}})}.$

The second main result is related to the exponential stabilization of (1.2).
For that, consider initially the following assumption

(1.10) $\lambda_{0}(x)\geq\gamma_{0}\quad\text{for a.e.}\quad x\in\mathbb{R},$

with some positive constant $\gamma_{0}.$ If the coefficient of the delay term
$\lambda$ satisfies the estimate
$\left\lVert\lambda\right\rVert_{\infty}<\gamma_{0},$ we can obtain the
exponential decay estimates for the ${\mathcal{E}}(t)$ associated to the
solution of the system (1.2), where

(1.11)
${\mathcal{E}}(t):={\mathcal{E}}(u(t))=\frac{1}{2}\int_{\mathbb{R}}u^{2}(x,t)dx+\frac{1}{2}\int_{t-\tau}^{t}\int_{\mathbb{R}}e^{-(t-s)}|\lambda(x)|u^{2}(x,s)\
dx\ ds.$

In that case, the delay effect is compensated by the undelayed damping term
(cf. [27]), and the following result holds:

######  Theorem 1.2.

Let $\lambda_{0},\lambda\in L^{\infty}(\mathbb{R})$ satisfying (1.10) and

(1.12) $\frac{e^{\tau}+1}{2}|\lambda(x)|\leq\gamma+\beta(x)\quad\text{for
a.e.}\quad x\in\mathbb{R}$

with

(1.13)
$0\leq\gamma<\gamma_{0}\quad\text{and}\quad\left\lVert\beta\right\rVert_{q}<\Big{(}\frac{\gamma_{0}-\gamma}{c_{q}}\Big{)}^{1-\frac{1}{2q}}.$

Here,

(1.14)
$c_{q}:=\Big{(}1-\frac{1}{2q}\Big{)}\Big{(}\frac{2}{q}\Big{)}^{\frac{1}{2q-1}}\quad\text{for}\quad
1\leq q<\infty.$

Thus, the solution of (1.2) is exponentially stable. Moreover, the solution
$u$ of (1.2) satisfies

(1.15) ${\mathcal{E}}(t)\leq C(u_{0},\tau)e^{-\nu t}.$

Here, $\nu$ is defined by

(1.16)
$\nu=\min\left\\{2\Big{(}\gamma_{0}-\gamma-\frac{2q-1}{2q}\Big{(}\frac{2}{q}\Big{)}^{\frac{1}{2q-1}}\|\beta\|_{q}^{\frac{2q}{2q-1}}\Big{)},1\right\\}$

and

(1.17)
$C(u_{0},\tau)=\frac{1}{2}\|u(0)\|^{2}_{2}+\int_{-\tau}^{0}e^{s}\|\lambda\|_{\infty}\|u(s)\|_{2}^{2}ds.$

After having restricted ourselves to the case where $\lambda_{0}$ is bounded
from below by a positive constant, the next issue is to extend our results to
the case where the coefficient of the undelayed feedback $\lambda_{0}$ is
indefinite. We have the following result:

######  Theorem 1.3.

Consider $\lambda_{0},\lambda\in L^{\infty}(\mathbb{R})$ satisfying (1.12) and

(1.18) $\lambda_{0}(x)\geq\gamma_{0}-\beta_{0}(x)\quad\text{for a.e.}\quad
x\in\mathbb{R},$

with $\beta_{0}$ satisfying

(1.19)
$\|\beta_{0}\|_{q}<\Big{(}\frac{\gamma_{0}}{c_{q}}\Big{)}^{1-\frac{1}{2q}}$

and

(1.20)
$0\leq\gamma<\gamma_{0}\quad\text{and}\quad\|\beta_{0}+\beta\|_{q}<\Big{(}\frac{\gamma_{0}-\gamma}{c_{q}}\Big{)}^{1-\frac{1}{2q}}.$

For every $u_{0}\in C([-\tau,0];L^{2}(\mathbb{R}))$, the system (1.2) has a
unique global mild solution. It is exponentially stable, that is,

${\mathcal{E}}(t)\leq C(u_{0},\tau)e^{-\tilde{\nu}t}$

with $C(u_{0},\tau)>0$ as in (1.17), and $\tilde{\nu}$ defined by

(1.21)
$\tilde{\nu}=\min\left\\{2\Big{(}\gamma_{0}-\gamma-\frac{2q-1}{2q}\Big{(}\frac{2}{q}\Big{)}^{\frac{1}{2q-1}}\|\beta+\beta_{0}\|_{q}^{\frac{2q}{2q-1}}\Big{)},1\right\\}.$

As a consequence of Theorem 1.3, we can find information on the solution of
the system (1.2) for any interval $[t,t+T]$, for $T>0$. The next result will
be crucial in getting the exponential stabilization in the space
$H^{2j+1}({\mathbb{R}})$.

######  Corollary 1.4.

Let $T>0$ and $u_{0}\in C([-\tau,0];L^{2}({\mathbb{R}}))$. Consider
$\lambda_{0}$ and $\lambda$ satisfying (1.12) and (1.18), respectively. Then,
there exists a nondecreasing continuous function
$\alpha_{0}:{\mathbb{R}}^{+}\rightarrow{\mathbb{R}}^{+}$ such that the
corresponding solution $u$ of problem (1.2) with $1\leq p<2j$ satisfies

(1.22) $\|u\|_{{\mathcal{B}}_{0,[t,t+T]}}\leq
C_{2T}\left\\{2C(u_{0},\tau)e^{-\nu
t}+\|\lambda\|_{\infty}\|u\|_{L^{1}(t-\tau,t;L^{2}(\mathbb{R}))}+\|\lambda\|_{\infty}^{1/2}\|u\|_{L^{2}(t-\tau,t;L^{2}(\mathbb{R}))}\right\\}$

where $C_{T}$ is given by (2.12) below and $C(u_{0},\tau)$ defined by (1.17).

Considering the general equation (1.2) in the $H^{s}$-norm, with
$s\in[0,2j+1]$ without the presence of the time-delay term, that is,
$\lambda=\tau=0$, the previous corollary helps us to prove the following
theorem on exponential decay:

######  Theorem 1.5.

Consider $\lambda_{0}\in L^{\infty}(\mathbb{R})$ satisfying (1.10) an (1.12).
For $\lambda=\tau=0$, the system (1.2) is exponentially stable. In other
words, there exist a time $T_{0}>0$ and a positive function $\eta(s)>0$ such
that for every $u_{0}\in H^{s}(\mathbb{R})$ with $s\in[0,2j+1]$, $1\leq p<2j$,
(1.2) has a unique global mild solution satisfying

(1.23)
$\|u(t)\|_{H^{s}({\mathbb{R}})}\leq\gamma(\|u_{0}\|_{2},T_{0})e^{-\eta(s)t},\quad\text{for
$t\geq T_{0}$}.$

Here, $\gamma:(0,\infty)\times{\mathbb{R}}^{+}\rightarrow{\mathbb{R}}^{+}$ is
a continuous function.

###  1.4. Comments and structure of the article

Let us give some general comments on our results.

  * 1.

It is important to point out that to prove Theorem 1.1 we use a method
introduced by Tartar [33] and adapted by Bona and Scott [8, Theorem 4.3]. This
method was used first to prove the global well-posedness of IVP for the KdV
equation on the whole line, and here, we adapt the spaces for our purpose.

  * 2.

Theorems 1.2 and 1.3 generalize the results in [21] for the general
differential operator

$Au:=(-1)^{j+1}\partial_{x}^{2j+1}u(x,t)+(-1)^{m}\partial_{x}^{2m}u(x,t),$

that is, considering appropriate values of $j$, we can recover the results in
[21]. Additionally to that, considering the real line, the results proved here
can be seen as extensions of [3, 2, 11, 12, 32].

  * 3.

Note that Theorem 1.5 extends (and recover) the previous results of [12, 14]
by proving the stabilization in ${\mathcal{B}}_{s,T}$ for $0\leq s\leq 2j+1$
with only a damping term, without the presence of the time-delay term.

  * 4.

The mains results of the paper are summarized in the following table:

General dispersive system | Well-posedness | Exponential stabilizition
---|---|---
With damping (indefined and localized) | ${\mathcal{B}}_{s,T}$ | $\|u(t)\|_{2}\leq C(u_{0},\tau)e^{-\eta t}$
\+ Time Delay | $0\leq s\leq 2j+1$ |
Only with damping | ${\mathcal{B}}_{s,T}$ | $\|u(t)\|_{H^{s}({\mathbb{R}})}\leq e^{-\eta t}\|u_{0}\|_{H^{s}({\mathbb{R}})}$
(indefined and localized) | $0\leq s\leq 2j+1$ | $0\leq s\leq 2j+1$

The structure of our paper is as follows. In Section 2 we analyze the well-
posedness of the system (1.2) when the initial data belongs to space
$C([-\tau,0]:H^{s}({\mathbb{R}}))$, where $s\in[0,2j+1]$ under some
assumptions of $\lambda_{0}$ and $\lambda$. Section 3 is devoted to the
exponential stabilization of the system (1.2) in $L^{2}({\mathbb{R}})$, that
is, to the proof of Theorems 1.2 and 1.3. In Section 4 we prove Theorem 1.5,
i.e., we establish the exponential stabilization of the system (1.2) without
the presence of time delay term ($\lambda=\tau=0$) in the space
$H^{s}({\mathbb{R}}))$, $s\in[0,2j+1]$. Finally, we conclude with some remarks
in Section 5.

##  2\. Well-posedness

In this section, we establish the well-posedness in $H^{s}({\mathbb{R}})$ with
$s\in[0,2j+1].$ Before presenting the proof of the Theorem 1.1, we establish
some well-posedness results in $L^{2}({\mathbb{R}})$ and
$H^{2j+1}({\mathbb{R}})$, respectively.

###  2.1. Linear system

For simplicity, consider $m=j\in\mathbb{N}$ in (1.2). Let us start studying
the following linear model associated with (1.2), namely

(2.1)
$\begin{cases}u_{t}(x,t)+(-1)^{j+1}\partial_{x}^{2j+1}u(x,t)+(-1)^{j}\partial_{x}^{2j}u(x,t)+\lambda_{0}u(x,t)\\\
\hskip 113.81102pt+\lambda
u(x,t-\tau)=0&\quad\text{in}\quad\mathbb{R}\times(0,\infty),\\\
u(x,s)=u_{0}(x,s)&\quad\text{in}\quad\mathbb{R}\times[-\tau,0].\end{cases}$

This section is devoted to proving the well-posedness via Semigroup theory and
so, with suitable conditions in the coefficients $\lambda$ and $\lambda_{0},$
to deduce the stabilization for the solutions associated to (2.1). We first
take a look at the properties of the following operator

(2.2)
$A_{\lambda_{0}}u:=-(-1)^{j+1}\partial_{x}^{2j+1}u(x,t){\color[rgb]{0,0,0}-}(-1)^{j}\partial_{x}^{2j}u(x,t)-\lambda_{0}u.$

The following well-posedness result can be proved.

######  Proposition 2.1.

If $\lambda_{0}\in L^{\infty}(\mathbb{R}),$ then the operator
$A_{\lambda_{0}}$ defined by (2.2) on
${\mathcal{D}}(A_{\lambda_{0}}):=H^{2j+1}(\mathbb{R})$ generates a strongly
continuous semigroup in the Hilbert space $H:=L^{2}(\mathbb{R}).$

###### Proof.

Note that $A_{\lambda_{0}}=A+\bar{A}_{\lambda_{0}}$, where
$Au=-(-1)^{j+1}\partial_{x}^{2j+1}u(x,t)-(-1)^{j}\partial_{x}^{2j}u(x,t)$ and
$\bar{A}_{\lambda_{0}}=-\lambda_{0}u$. Note that it suffices to observe that
$\bar{A}_{\lambda_{0}}$ is a bounded perturbation of the operator $A$ and to
prove that $A$ generates a strongly continuous semigroup in
$L^{2}(\mathbb{R}).$ Indeed, according to the Lumer-Phillips theorem it is
sufficient to check that $A$ is dissipative and that $I-A$ is onto. The
dissipativity follows by a direct computation: if $u_{0}\in
H^{2j+1}(\mathbb{R})$ is real-valued, then $u$ is also real-valued and

$\displaystyle(Au,u)_{H}$
$\displaystyle=\int_{-\infty}^{\infty}\left(-(-1)^{j+1}\partial_{x}^{2j+1}u{\color[rgb]{0,0,0}-}(-1)^{j}\partial_{x}^{2j}u\right)udx=-\int_{-\infty}^{\infty}(\partial^{j}_{x}u)^{2}dx\leqslant
0,$

since we have

$\int_{-\infty}^{\infty}(-1)^{j+1}(\partial_{x}^{2j+1}u)udx=0.$

Thanks to the fact that $\Re Av=A(\Re v)$ for all $v\in H^{2j+1}(\mathbb{R})$,
it follows that

$\Re(Au,u)_{H}=-\int_{-\infty}^{\infty}(\partial^{j}_{x}u)^{2}dx\leqslant 0$

for all $u_{0}\in H^{2j+1}(\mathbb{R})$.

It remains to show that for every $f\in L^{2}(\mathbb{R})$ there exists $u\in
H^{2j+1}(\mathbb{R})$ satisfying the equality

$-(-1)^{j+1}\partial_{x}^{2j+1}u(x,t){\color[rgb]{0,0,0}-}(-1)^{j}\partial_{x}^{2j}u(x,t)+u=f.$

Taking the Fourier transform it is equivalent to

$\hat{u}(\xi)\left(1-((-1)^{j+1}(i\xi)^{2j+1}){\color[rgb]{0,0,0}-}(-1)^{j}(i\xi)^{2j}\right)=\hat{f}(\xi)$

or equivalently

$\hat{u}(\xi)=\frac{\hat{f}(\xi)}{{\color[rgb]{0,0,0}1}-((-1)^{j+1}(i\xi)^{2j+1}){\color[rgb]{0,0,0}-}(-1)^{j}(i\xi)^{2j}}.$

This is possible by the fact that the denominator

$h(\xi):=1-((-1)^{j+1}(i\xi)^{2j+1}){\color[rgb]{0,0,0}-}(-1)^{j}(i\xi)^{2j}$

never vanishes. Since, moreover, $h(\xi)$ is a continuous function satisfying
$|h(\xi)|\rightarrow\infty$ as $|\xi|\rightarrow\infty,1/h$ is bounded, and
therefore the last equation has a unique solution $\hat{u}\in
L^{2}(\mathbb{R})$. Finally, since the function

$\frac{1+|\xi|+|\xi|^{2}+|\xi|^{3}+\cdots+|\xi|^{2j+1}}{\left|1-((-1)^{j+1}(i\xi)^{2j+1}){\color[rgb]{0,0,0}-}(-1)^{j}(i\xi)^{2j}\right|}$

tends to 1 as $|\xi|\rightarrow\infty$ and hence it is bounded by some
constant $M$ on $\mathbb{R}$, we conclude that

$\left|(i\xi)^{j}\hat{u}(\xi)\right|\leqslant M|\hat{f}(\xi)|,\quad
j\in\mathbb{N}.$

Since $\hat{f}\in L^{2}(\mathbb{R})$, this implies the regularity property
$u\in H^{2j+1}(\mathbb{R})$. ∎

Let us now use an iterative procedure (see e.g. [28]) and semigroup theory
(see e.g. [30]), to prove that (2.1) is well-posed.

######  Theorem 2.2.

If $\lambda_{0},\lambda\in L^{\infty}(\mathbb{R})$ and $u_{0}\in
C([-\tau,0];H)$, then there exists a unique solution $u\in
C([-\tau,+\infty);H)$ of the problem (2.1).

###### Proof.

Consider the interval $[0,\tau].$ Note that (2.1) can be seen as an
inhomogeneous Cauchy problem of the form

(2.3)
$\begin{cases}u_{t}(t)-A_{\lambda_{0}}u(t)=g_{0}(t)&\quad\text{in}\quad(0,\tau),\\\
u(0)=u_{0},\end{cases}$

with $g_{0}(t)=-\lambda u_{0}(t-\tau),$ for $t\in[0,\tau],$ which have unique
solution $u(\cdot)\in C([0,\tau),H).$ Now, considering the interval
$t\in[\tau,2\tau]$, problem (2.1) can be rewritten as

(2.4)
$\begin{cases}u_{t}(t)-A_{\lambda_{0}}u(t)=g_{1}(t)&\quad\text{in}\quad(\tau,2\tau),\\\
u(\tau)=u({\tau}_{-}),\end{cases}$

where $g_{1}(t)=-\lambda u(t-\tau).$ Thanks to the first step of the proof the
function $u(t)$, for $t\in[0,\tau]$, its known, thus $g_{1}(t)$ can be
considered as a known function for $t\in[\tau,2\tau].$ Therefore, this
analysis yields the existence of a solution $u(\cdot)\in C([0,2\tau],H).$ By a
bootstrap argument we get a solution $u\in C([0,\infty),H).$ ∎

###  2.2.  Non-homogeneous system

We are interested in extending the previous results for the nonlinear system
(1.2). In this way, we first consider the corresponding linear inhomogeneous
initial value problem

(2.5)
$\begin{cases}u_{t}(x,t)+(-1)^{j+1}\partial_{x}^{2j+1}u(x,t)+(-1)^{j}\partial_{x}^{2j}u(x,t)+\lambda_{0}u(x,t)\\\
\hskip 113.81102pt+\lambda
u(x,t-\tau)=f(x,t)&\quad\text{in}\quad\mathbb{R}\times(0,T),\\\
u(x,s)=u_{0}(x,s)&\quad\text{in}\quad\mathbb{R}\times[-\tau,0],\end{cases}$

for some $T>0.$

Consider the operator $A_{\lambda_{0}}$ defined by Proposition 2.1. So, we may
rewrite (2.5) in the following way

(2.6) $\begin{cases}u_{t}(x,t)+\lambda
u(x,t-\tau)=A_{\lambda_{0}}u(x,t)+f(x,t)&\quad\text{in}\quad\mathbb{R}\times(0,T),\\\
u(x,s)=u_{0}(x,s)&\quad\text{in}\quad\mathbb{R}\times[-\tau,0].\end{cases}$

As $A_{\lambda_{0}}$ generates a strongly continuous semigroup of contractions
in $L^{2}(\mathbb{R})$ by Proposition 2.1, for any given data $u_{0}\in
C([-\tau,0],H)$ and $f\in L^{1}(0,T;L^{2}(\mathbb{R})),$ problem (2.6) has a
unique mild solution $u\in C([-\tau,T];L^{2}(\mathbb{R})),$ satisfying the
following Duhamel’s formula

(2.7) $u(t)=S(t)u_{0}(0)-\int_{0}^{t}S(t-s)\lambda
u(s-\tau)\,ds+\int_{0}^{t}S(t-s)f(s)\,ds,\quad t\in[0,T].$

Therefore, we can prove that the mild solution of (2.6) depends continuously
on the initial data.

######  Proposition 2.3.

If $u_{0}\in C([-\tau,0],H)$ and $f\in L^{1}(0,T;L^{2}(\mathbb{R})),$ then the
solution of (2.6) satisfies the following estimate:

(2.8) $\|u(t)\|_{C([0,T];L^{2}(\mathbb{R}))}\leq
e^{\|\lambda\|_{\infty}T}\left(\|u(0)\|_{2}+\|f\|_{L^{1}(0,T;L^{2}(\mathbb{R}))}+\|\lambda\|_{\infty}\int_{-\tau}^{0}\|u(s)\|_{2}\,ds\right)$

and

(2.9) $\|u(t)\|_{C([-\tau,T];L^{2}(\mathbb{R}))}\leq
C\left(\|u_{0}\|_{C([-\tau,0];L^{2}(\mathbb{R}))}+\|f\|_{L^{1}(0,T;L^{2}(\mathbb{R}))}\right),$

where the positive constant $C$ is given by
$C=C(\|\lambda\|_{\infty},T,\tau)$.

###### Proof.

The Duhamel’s formula (2.7) give us that

$\displaystyle\|u(t)\|_{2}\leq$
$\displaystyle\|u(0)\|_{2}+\|\lambda\|_{\infty}\int_{0}^{t}\|u(s-\tau)\|_{2}\,ds+\|f\|_{L^{1}(0,T;L^{2}(\mathbb{R}))}$
$\displaystyle\leq$
$\displaystyle\|u(0)\|_{2}+\|f\|_{L^{1}(0,T;L^{2}(\mathbb{R}))}+\|\lambda\|_{\infty}\int_{-\tau}^{0}\|u(s)\|_{2}\,ds+\|\lambda\|_{\infty}\int_{0}^{t}\|u(s)\|_{2}\,ds$
$\displaystyle=$
$\displaystyle\color[rgb]{0,0,0}\|u(0)\|_{2}+\|f\|_{L^{1}(0,T;L^{2}(\mathbb{R}))}+\|\lambda\|_{\infty}\int_{-\tau}^{t}\|u(s)\|_{2}\,ds.$

Thus, the result follows by a direct application of Gronwall’s lemma. ∎

As a consequence of the previous inequality, we have the following
proposition.

######  Proposition 2.4.

Let $u_{0}\in C([-\tau,0],H)$ and $f\in L^{1}(0,T;L^{2}(\mathbb{R})),$ then
the solution of (2.6) belongs to ${\mathcal{B}}_{T}$ and the following
estimates holds true

(2.10) $\|u(t)\|_{C([-\tau,T];L^{2}(\mathbb{R}))}\leq
C(\|\lambda\|_{\infty},T,\tau)\left(\|u_{0}\|_{C([-\tau,0];L^{2}(\mathbb{R}))}+\|f\|_{L^{1}(0,T;L^{2}(\mathbb{R}))}\right)$

where $C=C(\|\lambda\|_{\infty},T,\tau)$ is a positive constant and

(2.11) $\|u\|_{{\mathcal{B}}_{T}}\leq
C_{T}\left\\{\|u(0)\|_{2}+\|f\|_{L^{1}(0,T;L^{2}(\mathbb{R}))}+\color[rgb]{0,0,0}\|\lambda\|_{\infty}\|u\|_{L^{1}(-\tau,0;L^{2}(\mathbb{R}))}+\|\lambda\|_{\infty}^{1/2}\|u\|_{L^{2}(-\tau,0;L^{2}(\mathbb{R}))}\right\\}$

where

(2.12)
$C_{T}=\sqrt{\frac{3}{2}}\left(1+e^{2\|\lambda\|_{\infty}T}\right)^{1/2}e^{(\|\lambda\|_{\infty}+\|\lambda_{0}\|_{\infty})T}.$

Moreover, we have that

(2.13)
$\displaystyle\begin{split}\frac{1}{2}\|u(t)\|_{2}^{2}+\int_{0}^{t}\|\partial^{j}_{x}u\|^{2}_{2}ds&+\int_{0}^{t}\int_{\mathbb{R}}\lambda_{0}u^{2}(x,s)dxds+\int_{0}^{t}\int_{\mathbb{R}}\lambda
u(x,s-\tau)u(x,s)dxds\\\
&=\frac{1}{2}\|u(0)\|_{2}^{2}+\int_{0}^{t}\int_{\mathbb{R}}f(x,s)u(x,s)dxds,\end{split}$

for all $t\in[0,T]$,

###### Proof.

Multiplying the equation (2.5) by $u$ and integrating by parts, taking into
account that

$\int_{\mathbb{R}}(-1)^{j+1}(\partial_{x}^{2j+1}u)udx=0$

and

$\int_{\mathbb{R}}(-1)^{j}\partial_{x}^{2j}u(x,t)udx=\int_{\mathbb{R}}(\partial^{j}_{x}u)^{2}dx,$

the relation (2.13) holds. Now, estimate (2.10) follows by (2.9).

Let us now prove (2.11). Thanks to (2.8) we infer that

$\displaystyle\begin{split}\|u(t)\|_{2}^{2}&+2\int_{0}^{t}\|\partial^{j}_{x}u\|_{2}^{2}ds\leq\|u(0)\|_{2}^{2}\\\
&+2\|f\|_{L^{1}(0,T;L^{2}(\mathbb{R}))}e^{\|\lambda\|_{\infty}T}\left(\|u(0)\|_{2}+\|f\|_{L^{1}(0,T;L^{2}(\mathbb{R}))}+\|\lambda\|_{\infty}\int_{-\tau}^{0}\|u(s)\|_{2}\,ds\right)\\\
&+2\|\lambda_{0}\|_{\infty}\int_{0}^{t}\|u(s)\|_{2}^{2}ds+\|\lambda\|_{\infty}\int_{0}^{t}\|u(s-\tau)\|_{2}^{2}ds+\|\lambda\|_{\infty}\int_{0}^{t}\|u(s)\|_{2}^{2}ds.\end{split}$

The previous inequality together with the following inequality

$\int_{0}^{t}\|u(s-\tau)\|_{2}^{2}\,ds\leq\int_{-\tau}^{0}\|u(s)\|_{2}^{2}\,ds+\int_{0}^{t}\|u(s)\|_{2}^{2}\,ds,$

ensures that

(2.14)
$\displaystyle\begin{split}\|u(t)\|_{2}^{2}&+2\int_{0}^{t}\|\partial^{j}_{x}u\|_{2}^{2}ds\leq\|u(0)\|_{2}^{2}+\|f\|_{L^{1}(0,T;L^{2}(\mathbb{R}))}^{2}\\\
&+e^{2\|\lambda\|_{\infty}T}\left(\|u(0)\|_{2}+\|f\|_{L^{1}(0,T;L^{2}(\mathbb{R}))}+\color[rgb]{0,0,0}\|\lambda\|_{\infty}\|u\|_{L^{1}(-\tau,0;L^{2}(\mathbb{R}))}\right)^{2}\\\
&+\|\lambda\|_{\infty}\|u\|_{L^{2}(-\tau,0;L^{2}(\mathbb{R}))}^{2}+2(\|\lambda\|_{\infty}+\|\lambda_{0}\|_{\infty})\int_{0}^{t}\|u\|^{2}_{2}ds.\end{split}$

From (2.14) we have

(2.15)
$\begin{split}\|u(t)\|_{2}^{2}&+2\int_{0}^{t}\|\partial^{j}_{x}u\|_{2}^{2}ds\leq
2(\|\lambda\|_{\infty}+\|\lambda_{0}\|_{\infty})\int_{0}^{t}\|u\|^{2}_{2}ds\\\
&+\Big{(}1+e^{2\|\lambda\|_{\infty}T}\Big{)}\left\\{\|u(0)\|_{2}+\|f\|_{L^{1}(0,T;L^{2}(\mathbb{R}))}\right.\\\
&\left.+\|\lambda\|_{\infty}\|u\|_{L^{1}(-\tau,0;L^{2}(\mathbb{R}))}+\|\lambda\|_{\infty}^{1/2}\|u\|_{L^{2}(-\tau,0;L^{2}(\mathbb{R}))}\right\\}^{2}.\end{split}$

An application of Gronwall’s Lemma gives the following

$\displaystyle\begin{split}\|u(t)\|_{2}^{2}&+2\int_{0}^{t}\|\partial^{j}_{x}u\|_{2}^{2}\,ds\leq\left(1+e^{2\|\lambda\|_{\infty}T}\right)e^{2(\|\lambda\|_{\infty}+\|\lambda_{0}\|_{\infty})T}\times\\\
&\times\Big{\\{}\|u(0)\|_{2}+\|f\|_{L^{1}(0,T;L^{2}(\mathbb{R}))}+\|\lambda\|_{\infty}\|u\|_{L^{1}(-\tau,0;L^{2}(\mathbb{R}))}+\|\lambda\|_{\infty}^{1/2}\|u\|_{L^{2}(-\tau,0;L^{2}(\mathbb{R}))}\Big{\\}}^{2},\end{split}$

thus

$\displaystyle\begin{split}\|u\|_{{\mathcal{B}}_{T}}^{2}\leq&\frac{3}{2}\left(1+e^{2\|\lambda\|_{\infty}T}\right)e^{2(\|\lambda\|_{\infty}+\|\lambda_{0}\|_{\infty})T}\times\\\
&\times\Big{\\{}\|u(0)\|_{2}+\|f\|_{L^{1}(0,T;L^{2}(\mathbb{R}))}+\color[rgb]{0,0,0}\|\lambda\|_{\infty}\|u\|_{L^{1}(-\tau,0;L^{2}(\mathbb{R}))}+\|\lambda\|_{\infty}^{1/2}\|u\|_{L^{2}(-\tau,0;L^{2}(\mathbb{R}))}\Big{\\}}^{2},\end{split}$

and so

$\displaystyle\begin{split}\|u\|_{{\mathcal{B}}_{T}}\leq&\sqrt{\frac{3}{2}}\left(1+e^{2\|\lambda\|_{\infty}T}\right)^{1/2}e^{(\|\lambda\|_{\infty}+\|\lambda_{0}\|_{\infty})T}\times\\\
&\times\Big{\\{}\|u(0)\|_{2}+\|f\|_{L^{1}(0,T;L^{2}(\mathbb{R}))}+\color[rgb]{0,0,0}\|\lambda\|_{\infty}\|u\|_{L^{1}(-\tau,0;L^{2}(\mathbb{R}))}+\|\lambda\|_{\infty}^{1/2}\|u\|_{L^{2}(-\tau,0;L^{2}(\mathbb{R}))}\Big{\\}}\end{split}$

showing (2.11) with $C_{T}$ defined by (2.12). ∎

###  2.3. Nonlinear estimates

In this subsection, we present some nonlinear estimates that will be used to
prove Theorem 1.1.

######  Lemma 2.5.

Let $1\leq p<2j$ with $j\geq 1$. Then, there exists a positive constant $C$,
such that, for any $T>0$ and $u,v\in B_{T}$, we have

$\|u^{p}v_{x}\|_{L^{1}(0,T;L^{2}({\mathbb{R}}))}\leq
2^{\frac{p}{2}}CT^{\frac{2j-p}{4j}}\|u\|_{\mathcal{B}_{T}}^{p}\|v\|_{\mathcal{B}_{T}}.$

###### Proof.

Recall that $H^{j}({\mathbb{R}})\hookrightarrow
H^{1}({\mathbb{R}})\hookrightarrow L^{\infty}({\mathbb{R}})$, for $j\geq 1$
and that (3.7) holds. On the other hand,

$\displaystyle\|u^{p}v_{x}\|_{L^{1}(0,T;L^{2}({\mathbb{R}}))}$
$\displaystyle\leq C\int_{0}^{T}\|u(t)\|_{\infty}^{p}\|v_{x}(t)\|_{2}dt$
$\displaystyle\leq
2^{\frac{p}{2}}C\int_{0}^{T}\|u(t)\|_{2}^{\frac{p}{2}}\|u_{x}(t)\|_{2}^{\frac{p}{2}}\|v_{x}(t)\|_{2}dt$
$\displaystyle\leq
2^{\frac{p}{2}}C\|u\|_{C([0,T];L^{2})}^{\frac{p}{2}}\int_{0}^{T}\|u_{x}(t)\|_{2}^{\frac{p}{2}}\|v_{x}(t)\|_{2}dt.$

Gagliardo-Nirenberg inequality says the following

(2.16) $\|\partial^{m}_{x}u\|_{2}\leq
C\|\partial^{j}_{x}u\|^{\frac{m}{j}}_{2}\|u\|_{2}^{1-\frac{m}{j}},\quad m\leq
j.$

Thus, by using (2.16) and Hölder inequality, we obtain

$\displaystyle\|u^{p}v_{x}\|_{L^{1}(0,T;L^{2}({\mathbb{R}}))}$
$\displaystyle\leq
2^{\frac{p}{2}}C\|u\|_{C([0,T];L^{2})}^{\frac{p}{2}}\int_{0}^{T}\|\partial_{x}^{j}u(t)\|_{2}^{\frac{p}{2j}}\|u(t)\|_{2}^{\frac{p}{2}\left(1-\frac{1}{j}\right)}\|v_{x}(t)\|_{2}dt$
$\displaystyle\leq
2^{\frac{p}{2}}C\|u\|_{C([0,T];L^{2})}^{\frac{p}{2}\left(2-\frac{1}{j}\right)}\left(\int_{0}^{T}\|\partial_{x}^{j}u(t)\|_{2}^{\frac{p}{2j}.\frac{4j}{p}}\right)^{\frac{p}{4j}}\left(\int_{0}^{T}\|v_{x}(t)\|_{2}^{2}dt\right)^{\frac{1}{2}}\left(\int_{0}^{T}dt\right)^{\frac{2j-p}{4j}}$
$\displaystyle\leq
2^{\frac{p}{2}}CT^{\frac{2j-p}{4j}}\|u\|_{C([0,T];L^{2})}^{\frac{p}{2}\left(2-\frac{1}{j}\right)}\|\partial_{x}^{j}u\|_{L^{2}([0,T];L^{2})}^{\frac{p}{2j}}\|\partial_{x}v\|_{L^{2}([0,T];L^{2})}$
$\displaystyle\leq
2^{\frac{p}{2}}CT^{\frac{2j-p}{4j}}\|u\|_{\mathcal{B}_{T}}^{\frac{p}{2}\left(2-\frac{1}{j}\right)}\|u\|_{\mathcal{B}_{T}}^{\frac{p}{2j}}\|v\|_{\mathcal{B}_{T}},$

proving the result. ∎

The next lemma gives the estimates for the nonlinear terms.

######  Lemma 2.6.

For any $T>0$, $1\leq p<2j$, $\lambda_{0}\in L^{\infty}({\mathbb{R}})$ and
$u,v,w\in\mathcal{B}_{T}$, we have

  1. (i)

$\|\lambda_{0}u\|_{L^{1}(0,T;L^{2}({\mathbb{R}}))}\leq
T^{\frac{1}{2}}\|\lambda_{0}\|_{\infty}\|u\|_{\mathcal{B}_{T}}$;

  2. (ii)

$\|uw_{x}\|_{L^{1}(0,T;L^{2}({\mathbb{R}}))}\leq
2^{\frac{1}{2}}T^{\frac{1}{4}}\|u\|_{\mathcal{B}_{T}}\|w\|_{\mathcal{B}_{T}}$;

  3. (iii)

$\|u|v|^{p-1}w_{x}\|_{L^{1}(0,T;L^{2}({\mathbb{R}}))}\leq
2^{\frac{p}{2}}T^{\frac{2j-p}{4j}}\|u\|_{\mathcal{B}_{T}}\|w\|_{\mathcal{B}_{T}}\|v\|_{\mathcal{B}_{T}}^{p-1}$;

  4. (iv)

The map $M:\mathcal{B}_{T}\rightarrow L^{1}(0,T;L^{2}({\mathbb{R}}))$, defined
by $Mu:=u^{p}u_{x}$, is locally Lipschitz continuous and satisfies

$\begin{split}\|Mu-
Mv\||_{L^{1}(0,T;L^{2}({\mathbb{R}}))}\leq&2^{\frac{p}{2}}T^{\frac{2j-p}{4j}}C\left(\|u\|_{\mathcal{B}_{T}}^{p}+\|u\|_{\mathcal{B}_{T}}\|v\|_{\mathcal{B}_{T}}^{p-1}+\|v\|_{\mathcal{B}_{T}}^{p}\right)\|u-v\|_{\mathcal{B}_{T}}\\\
&+2^{\frac{1}{2}}T^{\frac{1}{4}}C\|u\|_{\mathcal{B}_{T}}\|u-v\|_{\mathcal{B}_{T}},\end{split}$

for a positive constant $C$.

###### Proof.

Note that (i) follows using Hölder inequality. For (ii), observe that thanks
to (3.7), we have

$\displaystyle\|uw_{x}\|_{L^{1}(0,T;L^{2}({\mathbb{R}}))}$ $\displaystyle\leq
2^{\frac{1}{2}}T^{\frac{1}{4}}\|u\|_{\mathcal{B}_{T}}\|w\|_{\mathcal{B}_{T}}.$

Let us now prove (iii). Note that

$\displaystyle\|u|v|^{p-1}w_{x}\|_{L^{1}(0,T;L^{2}({\mathbb{R}}))}$
$\displaystyle\leq
2^{\frac{p}{2}}\|u\|_{\mathcal{B}_{T}}^{\frac{1}{2}}\|v\|_{\mathcal{B}_{T}}^{\frac{p-1}{2}}\int_{0}^{T}\|\partial_{x}^{j}u(t)\|_{2}^{\frac{1}{2j}}\|u(t)\|_{2}^{\frac{1}{2}\left(1-\frac{1}{j}\right)}\|\partial_{x}^{j}v(t)\|_{2}^{\frac{p-1}{2j}}\|v(t)\|_{2}^{\frac{p-1}{2}\left(1-\frac{1}{j}\right)}\|w_{x}(t)\|_{2}dt$
$\displaystyle\leq
2^{\frac{p}{2}}\|u\|_{\mathcal{B}_{T}}^{\frac{1}{2}}\|v\|_{\mathcal{B}_{T}}^{\frac{p-1}{2}}\|u\|_{\mathcal{B}_{T}}^{\frac{1}{2}\left(1-\frac{1}{j}\right)}\|v\|_{\mathcal{B}_{T}}^{\frac{p-1}{2}\left(1-\frac{1}{j}\right)}\int_{0}^{T}\|\partial_{x}^{j}u(t)\|_{2}^{\frac{1}{2j}}\|\partial_{x}^{j}v(t)\|_{2}^{\frac{p-1}{2j}}\|w_{x}(t)\|_{2}dt$
$\displaystyle=2^{\frac{p}{2}}\|u\|_{\mathcal{B}_{T}}^{\frac{1}{2}\left(2-\frac{1}{j}\right)}\|v\|_{\mathcal{B}_{T}}^{\frac{p-1}{2}\left(2-\frac{1}{j}\right)}\int_{0}^{T}\|\partial_{x}^{j}u(t)\|_{2}^{\frac{1}{2j}}\|\partial_{x}^{j}v(t)\|_{2}^{\frac{p-1}{2j}}\|w_{x}(t)\|_{2}dt.$

Thus, Hölder inequality ensures that

$\|u|v|^{p-1}w_{x}\|_{L^{1}(0,T;L^{2}({\mathbb{R}}))}\\\ \leq
2^{\frac{p}{2}}\|u\|_{\mathcal{B}_{T}}^{\frac{1}{2}\left(2-\frac{1}{j}\right)}\|v\|_{\mathcal{B}_{T}}^{\frac{p-1}{2}\left(2-\frac{1}{j}\right)}\left(\int_{0}^{T}\|\partial_{x}^{j}u\|_{2}^{2}dt\right)^{\frac{1}{4j}}\left(\int_{0}^{T}\|\partial_{x}^{j}v\|_{2}^{2}dt\right)^{\frac{p-1}{4j}}\left(\int_{0}^{T}\|w_{x}\|_{2}^{2}dt\right)^{\frac{1}{2}}T^{\frac{2j-p}{4j}}.$

The previous inequality gives

(2.17) $\displaystyle\|u|v|^{p-1}w_{x}\|_{L^{1}(0,T;L^{2}({\mathbb{R}}))}$
$\displaystyle\leq
2^{\frac{p}{2}}T^{\frac{2j-p}{4j}}\|u\|_{\mathcal{B}_{T}}^{\frac{1}{2}\left(2-\frac{1}{j}\right)}\|v\|_{\mathcal{B}_{T}}^{\frac{p-1}{2}\left(2-\frac{1}{j}\right)}\|u\|_{\mathcal{B}_{T}}^{\frac{1}{2j}}\|v\|_{\mathcal{B}_{T}}^{\frac{p-1}{2j}}\left(\int_{0}^{T}\|\partial_{x}^{j}w\|_{2}^{2}dt\right)^{\frac{1}{2}}$
$\displaystyle\leq
2^{\frac{p}{2}}T^{\frac{2j-p}{4j}}\|u\|_{\mathcal{B}_{T}}\|v\|_{\mathcal{B}_{T}}^{p-1}\|w\|_{\mathcal{B}_{T}},$

which allows us to get (iii). Finally, using the Mean Valued Theorem, (ii),
(iii) and Lemma 2.5, we have

$\displaystyle\|Mu-Mv\|_{L^{1}(0,T;L^{2})}\leq$ $\displaystyle
C\|(1+|u|^{p-1}+|v|^{p-1})|u-v|u_{x}\|_{L^{1}(0,T;L^{2})}+\|v^{p}(u-v)_{x}\|_{L^{1}(0,T;L^{2})}$
$\displaystyle\leq$ $\displaystyle
C\left\\{2^{\frac{1}{2}}T^{\frac{1}{4}}\|u-v\|_{\mathcal{B}_{T}}\|u\|_{\mathcal{B}_{T}}+2^{\frac{p}{2}}T^{\frac{2j-p}{4j}}\|u-v\|_{\mathcal{B}_{T}}\|u\|_{\mathcal{B}_{T}}^{p}\right.$
$\displaystyle\left.+2^{\frac{p}{2}}T^{\frac{2j-p}{4j}}\|u-v\|_{\mathcal{B}_{T}}\|u\|_{\mathcal{B}_{T}}\|v\|_{\mathcal{B}_{T}}^{p-1}+2^{\frac{p}{2}}T^{\frac{2-p}{4}}\|u-v\|_{\mathcal{B}_{T}}\|v\|_{\mathcal{B}_{T}}^{p}\right\\},$

and (iv) holds. ∎

###  2.4. Nonlinear system

We are in position now to consider the nonlinear model (1.2), with $u_{0}\in
C([-\tau,0];L^{2}(\mathbb{R})).$ Before presenting it, let us introduce the
following definitions

######  Definition 2.7.

A mild solution of (1.2) will be a function $u\in{\mathcal{B}}_{T},$ $T>0,$
which satisfies

$\displaystyle u(t)$ $\displaystyle=S(t)u_{0}(0)-\int_{0}^{t}S(t-s)\lambda
u_{0}(s-\tau)ds-\int_{0}^{t}S(t-s)u^{p}(s)\partial_{x}u(s)ds,\quad t\in[0,T].$

A global mild solution of (1.2) will be a function $u:[0,\infty)\rightarrow
H^{1}(\mathbb{R})$ whose restriction to every bounded interval $[0,T]$ is a
mild solution of (1.2).

####  2.4.1. Well-posedness in $L^{2}({{\mathbb{R}}})$

With these definitions in hand and the previous estimates, the following
result gives the local well-posedness for the higher order dispersive equation
and a priori estimate for the solutions of (1.2).

######  Proposition 2.8.

Let $1\leq p<2j$ with $j\geq 1$, and $\lambda_{0},\lambda\in
L^{\infty}({\mathbb{R}})$. For $u_{0}\in L^{2}({\mathbb{R}})$, there exist
$T>0$ and a unique mild solution $u\in\mathcal{B}_{T}$ of (1.2), such that

(2.18) $\|u\|_{{\mathcal{B}}_{T}}\leq
C_{T}\left\\{\|u(0)\|_{2}+\color[rgb]{0,0,0}\|\lambda\|_{\infty}\|u_{0}\|_{L^{1}(-\tau,0;L^{2}(\mathbb{R}))}+\|\lambda\|_{\infty}^{1/2}\|u_{0}\|_{L^{2}(-\tau,0;L^{2}(\mathbb{R}))}\right\\}.$

Here, $C_{T}$ is given by (2.12).

###### Proof.

Let $T>0$ be determined later. For each $u\in\mathcal{B}_{T}$ consider the
problem

(2.19) $\begin{cases}v_{t}=A_{\lambda_{0}}v-\lambda
v(x,t-\tau)-Mu&\quad\text{in}\quad\mathbb{R}\times(0,T),\\\
v(x,s)=u_{0}(x,s)&\quad\text{in}\quad\mathbb{R}\times[-\tau,0].\end{cases}$

Since $A_{\lambda_{0}}$ generates a strongly continuous semigroup
$\\{S(t)\\}_{t\geq 0}$ of contractions in $L^{2}({\mathbb{R}})$. Proposition
2.4 allows us to conclude that (2.19) has a unique mild solution $v\in
B_{0,T}$, such that

(2.20) $\|v\|_{{\mathcal{B}}_{T}}\leq
C_{T}\left\\{\|u_{0}\|_{2}+\|Mu\|_{L^{1}(0,T;L^{2}(\mathbb{R}))}+\|\lambda\|_{\infty}\|u_{0}\|_{L^{1}(-\tau,0;L^{2}(\mathbb{R}))}+\|\lambda\|_{\infty}^{1/2}\|u_{0}\|_{L^{2}(-\tau,0;L^{2}(\mathbb{R}))}\right\\}$

where $C_{T}$ is given by (2.12). Moreover, we have that

$\displaystyle\begin{split}\frac{1}{2}\|v(t)\|_{2}^{2}+\int_{0}^{t}\|\partial^{j}_{x}v\|^{2}_{2}ds&+\int_{0}^{t}\int_{\mathbb{R}}\lambda_{0}v^{2}(x,s)dxds+\int_{0}^{t}\int_{\mathbb{R}}\lambda
v(x,s-\tau)v(x,s)dxds\\\
&=\frac{1}{2}\|u_{0}\|_{2}^{2}+\int_{0}^{t}\int_{\mathbb{R}}Mu(x,s)v(x,s)dxds.\end{split}$

Thus, we can define the operator
$\Gamma:\mathcal{B}_{T}\longrightarrow\mathcal{B}_{T}$ given by $\Gamma(u)=v$.
Thanks to the Lemma 2.5 and (2.20), we have

$\|\Gamma u\|_{\mathcal{B}_{T}}\leq
C_{T}\\{\|u_{0}\|_{2}+2^{p/2}CT^{\frac{2j-p}{4j}}\|u\|_{\mathcal{B}_{T}}^{p+1}+\color[rgb]{0,0,0}\|\lambda\|_{\infty}\|u_{0}\|_{L^{1}(-\tau,0;L^{2}(\mathbb{R}))}+\|\lambda\|_{\infty}^{1/2}\|u_{0}\|_{L^{2}(-\tau,0;L^{2}(\mathbb{R}))}\\},$

and, for $u\in B_{R}(0):=\\{u\in\mathcal{B}_{T}:\|u\|_{\mathcal{B}_{T}}\leq
R\\}$, it follows that

$\|\Gamma u\|_{\mathcal{B}_{T}}\leq
C_{T}\\{\|u_{0}\|_{2}+2^{p/2}CT^{\frac{2j-p}{4j}}R^{p+1}+\color[rgb]{0,0,0}\|\lambda\|_{\infty}\|u_{0}\|_{L^{1}(-\tau,0;L^{2}(\mathbb{R}))}+\|\lambda\|_{\infty}^{1/2}\|u_{0}\|_{L^{2}(-\tau,0;L^{2}(\mathbb{R}))}\\}.$

Choosing

$R=2C_{T}\left(\|u_{0}\|_{2}+\left(\|\lambda\|_{\infty}\|u_{0}\|_{L^{1}(-\tau,0;L^{2}(\mathbb{R}))}+\|\lambda\|_{\infty}^{1/2}\|u_{0}\|_{L^{2}(-\tau,0;L^{2}(\mathbb{R}))}\right)\|u_{0}\|_{L^{2}(-\tau,0;L^{2}(\mathbb{R}))}\right),$

we obtain the following estimate

$\|\Gamma u\|_{0,T}\leq\left(K_{1}+\frac{1}{2}\right)R,$

where $K_{1}=K_{1}(T)=2^{p/2}C_{T}CT^{\frac{2j-p}{4j}}R^{p}.$ On the other
hand, note that $\Gamma u-\Gamma w$ is solutions of

$\begin{cases}v_{t}=A_{\lambda_{0}}v-(Mu-
Mv)&\quad\text{in}\quad\mathbb{R}\times(0,T),\\\
v(x,s)=0&\quad\text{in}\quad\mathbb{R}\times[-\tau,0].\end{cases}$

We are now in a position to prove that $\Gamma$ has a unique fixed point. To
do that, note that thanks to Proposition 2.4, we have

$\displaystyle\|\Gamma u-\Gamma w\|_{\mathcal{B}_{T}}$ $\displaystyle\leq
C_{T}\|Mu-Mw\|_{L^{1}(0,T;L^{2})}.$

Lemma 2.6, precisely estimate (iv), allows us to conclude that

$\displaystyle\|\Gamma u-\Gamma w\|_{\mathcal{B}_{T}}\leq
C_{T}C\left\\{2^{\frac{p}{2}}T^{\frac{2j-p}{4j}}\left(\|u\|_{\mathcal{B}_{T}}^{p}+\|u\|_{\mathcal{B}_{T}}\|w\|_{\mathcal{B}_{T}}^{p-1}+\|w\|_{\mathcal{B}_{T}}^{p}\right)+2^{\frac{1}{2}}T^{\frac{1}{4}}\|u\|_{\mathcal{B}_{T}}\right\\}\|u-w\|_{\mathcal{B}_{T}}.$

Suppose that $u,w\in B_{R}(0)$ defined above. Then,

$\displaystyle\|\Gamma u-\Gamma w\|_{\mathcal{B}_{T}}$ $\displaystyle\leq
K_{2}\|u-w\|_{\mathcal{B}_{T}},$

where
$K_{2}=K_{2}(T)=C_{T}C\\{2^{\frac{1}{2}}T^{\frac{1}{4}}R+3(2^{\frac{p}{2}})T^{\frac{2j-p}{4j}}R^{p}\\}$.
Since $K_{1}\leq K_{2}$, we can choose $T>0$ to obtain $K_{2}<\frac{1}{2}$,
$\|\Gamma u\|_{\mathcal{B}_{T}}\leq R$ and

$\|\Gamma u-\Gamma
w\|_{\mathcal{B}_{T}}<\frac{1}{2}\|u-w\|_{\mathcal{B}_{T}},$

for all $u,w\in B_{R}(0)\subset B_{0,T}$. Hence,
$\Gamma:B_{R}(0)\longrightarrow B_{R}(0)$ is a contraction and, by Banach
fixed point theorem, we obtain a unique $u\in B_{R}(0)$, such that
$\Gamma(u)=u$ and consequently, the local well-posedness result for
$0<T\leq\tau$ small enough to the system. Thus, $u$ is a unique local mild
solution to the problem, and estimate (2.8) holds. ∎

We are now in a position to present the main result of this subsection.

######  Theorem 2.9.

Consider $\lambda_{0},\lambda\in L^{\infty}(\mathbb{R})$. Therefore, the
system (1.2) admits a unique global mild solution for every initial data
$u_{0}\in C([-\tau,0];L^{2}(\mathbb{R}))$ satisfying

(2.21)
$\frac{1}{2}\|u(t)\|_{2}^{2}+\int_{0}^{t}\|u_{x}\|^{2}_{2}\,ds+\int_{0}^{t}\int_{\mathbb{R}}\lambda_{0}u^{2}(x,s)dxds+\int_{0}^{t}\int_{\mathbb{R}}\lambda
u(x,s-\tau)u(x,s)dxds=\frac{1}{2}\|u(0)\|_{2}^{2},$

for all $t\geq 0$. Moreover, there exists a non-decreasing continuous function
$\beta_{0}:{\mathbb{R}}_{+}\rightarrow{\mathbb{R}}_{+}$, such that the
solution $u$ (1.2) satisfies

(2.22) $\|u\|_{{\mathcal{B}}_{T}}\leq
C_{T}\left\\{\|u(0)\|_{2}+\color[rgb]{0,0,0}\|\lambda\|_{\infty}\|u_{0}\|_{L^{1}(-\tau,0;L^{2}(\mathbb{R}))}+\|\lambda\|_{\infty}^{1/2}\|u_{0}\|_{L^{2}(-\tau,0;L^{2}(\mathbb{R}))}\right\\}$

and

(2.23)
$\|u\|_{{\mathcal{B}}_{T}}\leq\beta_{0}(\|u_{0}\|_{2})\|u_{0}\|_{C([-\tau,0];L^{2}(\mathbb{R}))},$

with
$\beta_{0}=C_{T}\left(1+\left(\|\lambda\|_{\infty}\tau^{1/2}+\|\lambda\|_{\infty}^{1/2}\right)\right)$
and $C_{T}$ given by (2.12).

###### Proof.

To prove the global well-posedness we need to prove that the norms of the
solutions of (1.2) remain bounded in the existence time interval. To do that,
let us consider the functional ${\mathcal{E}}(\cdot)$, previously defined by
(3.2). Taking the time derivative in time of this function, we have that

$\displaystyle\begin{split}\frac{d{\mathcal{E}}}{dt}(t)=&\int_{\mathbb{R}}u(t)((-1)^{j}\partial^{2j}_{x}u(t)-\lambda_{0}u(t)-\lambda
u(t-\tau)+u^{p}(t)u_{x}(t))dx+\frac{1}{2}\int_{\mathbb{R}}|\lambda|u^{2}(t)dx\\\
&-\frac{1}{2}e^{-\tau}\int_{\mathbb{R}}|\lambda|u^{2}(t-\tau)dx-\frac{1}{2}\int_{t-\tau}^{t}\int_{\mathbb{R}}e^{-(t-s)}|\lambda|u^{2}(x,s)dxds.\end{split}$

Integrating by parts, using the Young inequality and taking into account the
hypothesis (1.10), (1.12), and (1.13) we get that

$\displaystyle\begin{split}\frac{d{\mathcal{E}}}{dt}(t)\leq&-\int_{\mathbb{R}}(\partial^{j}_{x}u)^{2}(t)dx-\gamma_{0}\int_{\mathbb{R}}u^{2}(t)dx+\frac{e^{\tau}+1}{2}\int_{\mathbb{R}}|\lambda(x)|u^{2}(t)dx\\\
&-\frac{1}{2}\int_{t-\tau}^{t}\int_{\mathbb{R}}e^{-(t-s)}|\lambda|u^{2}(x,s)dxds\\\
\leq&-\int_{\mathbb{R}}(\partial^{j}_{x}u)(t)dx-(\gamma_{0}-\gamma)\int_{\mathbb{R}}u^{2}(t)dx+\int_{\mathbb{R}}\beta(x)u^{2}(t)dx\leq
0.\end{split}$

Here, we have used that the third integral in the previous inequality can be
handled as in the proof of Theorem 3.1. Therefore, this inequality ensures
that $\|u(t)\|_{2}$ remains bounded for $t\in[0,T].$

Finally, thanks to the estimate (2.21), we deduce that
$\|u\|_{\mathcal{B}_{T}}$ remains bounded for $t\in[0,T]$, and so, the local
solution $u$ given by Proposition 2.8 can be extended on $[0,\tau].$ Now, once
we have a solution $u\in{\mathcal{B}}_{\tau}$ we can apply the same arguments
as we did on Theorem 2.2 to prove the existence of a global mild solution of
(1.2). Finally, note that the proof of Proposition 2.8 guarantees the function
$\beta_{0}$ is given by

$\displaystyle\beta_{0}(s)=C_{T}\left(1+\left(\|\lambda\|_{\infty}\tau^{1/2}+\|\lambda\|_{\infty}^{1/2}\right)\right),$

showing the result. ∎

####  2.4.2. Well-posedness in $H^{2j+1}({\mathbb{R}})$

We will analyze the well-posedness in ${\mathcal{B}}_{2j+1,T}$, with $1\leq
p<2j$ with $j\geq 1$. To do that, let us first consider the following
linearized problem

(2.24)
$\begin{cases}v_{t}+(-1)^{j+1}\partial_{x}^{2j+1}v+(-1)^{m}\partial_{x}^{2m}v+\partial_{x}(u^{p}v)\\\
\hskip
113.81102pt+\lambda_{0}(x)v+\lambda(x)v(x,t-\tau)=0&\quad\text{in}\quad\mathbb{R}\times(0,\infty),\\\
v(x,s)=v_{0}(x,s)&\quad\text{in}\quad\mathbb{R}\times[-\tau,0].\end{cases}$

Then, we can establish the following proposition:

######  Proposition 2.10.

For $T>0$, $\lambda_{0},\lambda\in L^{\infty}({\mathbb{R}})$, and
$u\in{\mathcal{B}}_{T}$, if $v_{0}\in C([-\tau,0];L^{2}({\mathbb{R}}))$, then
system (2.24) admits a unique solution $v\in B_{0,T}$, such that

(2.25) $\|v(t)\|_{C([-\tau,T];L^{2}(\mathbb{R}))}\leq
C(\|\lambda\|_{\infty},T,\tau)\left(\|v_{0}\|_{C([-\tau,0];L^{2}(\mathbb{R}))}+(p+1)\|u\|_{{\mathcal{B}}_{T}}^{p}\|v\|_{{\mathcal{B}}_{T}}\right),$

where $C(\|\lambda\|_{\infty},T,\tau)$ is a positive constant and

(2.26) $\color[rgb]{0,0,0}\|v\|_{{\mathcal{B}}_{T}}\leq
C_{T}\left\\{\|v(0)\|_{2}+\|u\|_{{\mathcal{B}}_{T}}^{p}+\|\lambda\|_{\infty}\|v_{0}\|_{L^{1}(-\tau,0;L^{2}(\mathbb{R}))}+\|\lambda\|_{\infty}^{1/2}\|v_{0}\|_{L^{2}(-\tau,0;L^{2}(\mathbb{R}))}\right\\}$

and

(2.27)
$\color[rgb]{0,0,0}\|v\|_{{\mathcal{B}}_{T}}\leq\sigma(\|u\|_{{\mathcal{B}}_{T}},T)\left\\{\|v(0)\|_{2}+\|\lambda\|_{\infty}\|v_{0}\|_{L^{1}(-\tau,0;L^{2}(\mathbb{R}))}+\|\lambda\|_{\infty}^{1/2}\|v_{0}\|_{L^{2}(-\tau,0;L^{2}(\mathbb{R}))}\right\\}.$

Here, $1\leq p<2j$, with $j\leq 1$, and
$\sigma:{\mathbb{R}}^{+}\times{\mathbb{R}}^{+}\rightarrow{\mathbb{R}}^{+}$ is
a non-decreasing continuous function.

###### Proof.

The existence of a solution follows the same steps as done in Proposition 2.8
and Theorem 2.9, so we will omit it. Let us prove inequality (2.27). Thanks to
(2.10) and (2.11), we deduce (2.25) and

$\|v\|_{{\mathcal{B}}_{T}}\leq
C_{T}\left\\{\|v(0)\|_{2}+\|(u^{p}v)_{x}\|_{L^{1}(0,T;L^{2}(\mathbb{R}))}+\|\lambda\|_{\infty}\|v_{0}\|_{L^{1}(-\tau,0;L^{2}(\mathbb{R}))}+\|\lambda\|_{\infty}^{1/2}\|v_{0}\|_{L^{2}(-\tau,0;L^{2}(\mathbb{R}))}\right\\}.$

Lemma 2.6 and Young inequality imply that there exist a constant $C>0$ such
that

$\|v(t)\|_{C([-\tau,T];L^{2}(\mathbb{R}))}\leq
e^{\|\lambda\|_{\infty}T}\left(\|v(0)\|_{2}+(p+1)\|u\|_{{\mathcal{B}}_{T}}^{p}\|v\|_{{\mathcal{B}}_{T}}\right)$

and

$\|v\|_{{\mathcal{B}}_{T}}\leq
CC_{T}\left\\{\|v(0)\|_{2}+\|u\|_{{\mathcal{B}}_{T}}^{p}+\|\lambda\|_{\infty}\|v_{0}\|_{L^{1}(-\tau,0;L^{2}(\mathbb{R}))}+\|\lambda\|_{\infty}^{1/2}\|v_{0}\|_{L^{2}(-\tau,0;L^{2}(\mathbb{R}))}\right\\}.$

On the other hand, following the same ideas as done in Proposition 2.4, from
(2.15) we infer that

$\displaystyle\begin{split}\|v(t)\|_{2}^{2}&+2\int_{0}^{t}\|\partial^{j}_{x}v\|_{2}^{2}ds\leq
2(\|\lambda\|_{\infty}+\|\lambda_{0}\|_{\infty})\int_{0}^{t}\|v\|^{2}_{2}ds\\\
&+\Big{(}1+e^{2\|\lambda\|_{\infty}T}\Big{)}\Big{\\{}\|v(0)\|_{2}+\|\partial_{x}(u^{p}v)v\|_{L^{1}(0,t;L^{2}(\mathbb{R}))}+\color[rgb]{0,0,0}\|\lambda\|_{\infty}\|v\|_{L^{1}(-\tau,0;L^{2}(\mathbb{R}))}\\\
&+\|\lambda\|_{\infty}^{1/2}\|v\|_{L^{2}(-\tau,0;L^{2}(\mathbb{R}))}\Big{\\}}^{2}.\end{split}$

Gagliardo-Nerinberg inequality ensures that

$\displaystyle\|\partial_{x}(u^{p}v)v\|_{L^{1}(0,t;L^{2}(\mathbb{R}))}$
$\displaystyle=\|u^{p}v_{x}v\|_{L^{1}(0,t;L^{2}(\mathbb{R}))}\leq\int_{0}^{t}\|u^{p}(s)v_{x}(s)v(s)\|_{2}ds$
$\displaystyle\leq\|u\|_{{\mathcal{B}}_{T}}^{p}\int_{0}^{t}\|v_{x}(s)\|_{2}\|v(s)\|_{\infty}ds\leq
2^{\frac{1}{2}}\|u\|_{{\mathcal{B}}_{T}}^{p}\int_{0}^{t}\|v_{x}(s)\|^{\frac{3}{2}}_{2}\|v(s)\|_{2}^{\frac{1}{2}}ds$
$\displaystyle\leq
2^{\frac{1}{2}}C\|u\|_{{\mathcal{B}}_{T}}^{p}\int_{0}^{t}\|v(s)\|^{\frac{3}{2}\left(1-\frac{1}{j}\right)}_{2}\|\partial_{x}^{j}v(s)\|^{\frac{3}{2j}}_{2}\|v(s)\|_{2}^{\frac{1}{2}}ds$
$\displaystyle=2^{\frac{1}{2}}C\|u\|_{{\mathcal{B}}_{T}}^{p}\int_{0}^{t}\|v(s)\|^{\frac{4j-3}{2j}}_{2}\|\partial_{x}^{j}v(s)\|^{\frac{3}{2j}}_{2}ds$

and using young inequality, we have that

$\displaystyle\int_{0}^{t}\|v(s)\|^{\frac{4j-3}{2j}}_{2}\|\partial_{x}^{j}v(s)\|^{\frac{3}{2j}}_{2}ds\leq\frac{(4j-3)}{4j\eta^{\frac{4j}{4j-3}}}\int_{0}^{t}\|v(s)\|^{2}_{2}ds+\frac{3\eta^{\frac{4j}{3}}}{4j}\int_{0}^{t}\|\partial_{x}^{j}v(s)\|^{2}_{2}ds$

where
$\eta=\left[\frac{4j}{3\sqrt{2}C(1+e^{2\|\lambda\|_{\infty}T})\|u\|^{p}_{{\mathcal{B}}_{T}}}\right]^{\frac{3}{4j}}$.
Hence, it yields that

$\displaystyle\begin{split}\|v(t)\|_{2}^{2}+\int_{0}^{t}\|\partial^{j}_{x}v\|_{2}^{2}ds\leq&\rho(\|u\|_{{\mathcal{B}}_{T}})\int_{0}^{t}\|v(s)\|^{2}_{2}ds+\Big{(}1+e^{2\|\lambda\|_{\infty}T}\Big{)}\Big{\\{}\|v(0)\|_{2}+\|\lambda\|_{\infty}\|v\|_{L^{1}(-\tau,0;L^{2}(\mathbb{R}))}\\\
&+\|\lambda\|_{\infty}^{1/2}\|v\|_{L^{2}(-\tau,0;L^{2}(\mathbb{R}))}\Big{\\}}^{2},\end{split}$

where

$\begin{split}\rho(\|u\|_{{\mathcal{B}}_{T}})=&2(\|\lambda\|_{\infty}+\|\lambda_{0}\|_{\infty})\\\
&+(1+e^{2\|\lambda\|_{\infty}T})2^{\frac{1}{2}}C\|u\|_{{\mathcal{B}}_{T}}^{p}\frac{(4j-3)}{4j}\left[\frac{4j}{3\sqrt{2}C(1+e^{2\|\lambda\|_{\infty}T})\|u\|^{p}_{{\mathcal{B}}_{T}}}\right]^{-\frac{3}{4j-3}}\\\
=&2(\|\lambda\|_{\infty}+\|\lambda_{0}\|_{\infty})\\\
&+(4j-3)\left[\frac{3\sqrt{2}C(1+e^{2\|\lambda\|_{\infty}T})\|u\|^{p}_{{\mathcal{B}}_{T}}}{4j}\right]^{\frac{4j}{4j-3}}.\end{split}$

Finally, we have that

$\displaystyle\begin{split}\|v(t)\|_{2}^{2}+\int_{0}^{t}\|\partial^{j}_{x}v\|_{2}^{2}ds\leq&\rho(\|u\|_{{\mathcal{B}}_{T}})\int_{0}^{t}\left(\|v(s)\|^{2}_{2}+\int_{0}^{s}\|\partial^{j}_{x}v(t)\|_{2}^{2}dr\right)ds\\\
&+\Big{(}1+e^{2\|\lambda\|_{\infty}T}\Big{)}\Big{\\{}\|v(0)\|_{2}+\|\lambda\|_{\infty}\|v\|_{L^{1}(-\tau,0;L^{2}(\mathbb{R}))}\\\
&+\|\lambda\|_{\infty}^{1/2}\|v\|_{L^{2}(-\tau,0;L^{2}(\mathbb{R}))}\Big{\\}}^{2}.\end{split}$

Employing Gronwall’s inequality, we conclude

$\displaystyle\begin{split}\|v(t)\|_{2}^{2}+\int_{0}^{t}\|\partial^{j}_{x}v\|_{2}^{2}ds\leq&\Big{(}1+e^{2\|\lambda\|_{\infty}T}\Big{)}e^{\rho(\|u\|_{{\mathcal{B}}_{T}})t}\Big{\\{}\|v(0)\|_{2}+\|\lambda\|_{\infty}\|v\|_{L^{1}(-\tau,0;L^{2}(\mathbb{R}))}\\\
&+\|\lambda\|_{\infty}^{1/2}\|v\|_{L^{2}(-\tau,0;L^{2}(\mathbb{R}))}\Big{\\}}^{2}.\end{split}$

Estimate (2.27) follows directly from the above inequality. ∎

We now give the existence of a solution to the problem (1.2).

######  Theorem 2.11.

Let $T>0$, $\lambda_{0},\lambda\in H^{j}({\mathbb{R}})$ and $1\leq p<2j$, with
$j\geq 1$. For $u_{0}(x,t-\tau)\in C([-\tau,0];H^{2j+1}({\mathbb{R}}))$, there
exists a unique mild solution $u\in\mathcal{B}_{2j+1,T}$ of (1.2) such that

$\|u\|_{{\mathcal{B}}_{2j+1,T}}\leq\beta_{2j+1}(\|u_{0}\|_{2})\|u_{0}\|_{C([-\tau,0];H^{2j+1}({\mathbb{R}}))},$

where $\beta_{2j+1}:{\mathbb{R}}_{+}\rightarrow{\mathbb{R}}_{+}$ is a non-
decreasing continuous function.

###### Proof.

We will split the proof into several steps.

Step 1: $u\in L^{2}(0,T;H^{2j+1}({\mathbb{R}})).$

Since $u_{0}\in H^{2j+1}({\mathbb{R}})\hookrightarrow L^{2}({\mathbb{R}})$, by
Theorem 2.9, there exists a unique solution $u\in{\mathcal{B}}_{T}$, such that

(2.28)
$\|u\|_{{\mathcal{B}}_{T}}\leq\beta_{0}(\|u_{0}\|_{2})\|u_{0}\|_{C([-\tau,0];L^{2}(\mathbb{R}))}.$

We will show that $u\in{\mathcal{B}}_{2j+1,T}$. Let $v=u_{t}$. Then, $v$
solves the problem

$\left\\{\begin{tabular}[]{l
l}$v_{t}+(-1)^{j+1}\partial_{x}^{2j+1}v+(-1)^{m}\partial_{x}^{2m}v+\partial_{x}(u^{p}v)+\lambda_{0}(x)v+\lambda(x)v(x,t-\tau)=0$&in
${\mathbb{R}}\times(0,\infty)$,\\\ $v(x,0)=v_{0}(x)$&in ${\mathbb{R}}$,\\\
$v(x,s)=v_{0}(x,s)$&in ${\mathbb{R}}\times(-\tau,0)$,\end{tabular}\right.$

where
$v_{0}(x)=-(-1)^{j+1}\partial_{x}^{2j+1}u_{0}+(-1)^{m}\partial_{x}^{2m}u_{0}-\frac{1}{p+1}\partial_{x}(u_{0}^{p+1})-\lambda_{0}(x)u_{0}-\lambda(x)u_{0}(x,-\tau)$.
We can bound $v_{0}$ as follows:

$\displaystyle\|v_{0}\|_{2}$
$\displaystyle\leq\|\partial_{x}^{2j+1}u_{0}\|_{2}+\|\partial_{x}^{2m}u_{0}\|_{2}+\|u_{0}^{p}\partial_{x}u_{0}\|_{2}+\|\lambda_{0}u_{0}\|_{2}+\|\lambda(x)u_{0}(-\tau)\|_{2}$
$\displaystyle\leq(1+\|\lambda_{0}\|_{L^{\infty}({\mathbb{R}})})\|u_{0}\|_{H^{2j+1}({\mathbb{R}})}+\|u_{0}\|^{\frac{p}{2}}_{\infty}\|\partial_{x}u_{0}\|_{2}+\|\lambda\|_{\infty}\|u_{0}(\cdot,-\tau)\|_{2}$
$\displaystyle\leq(1+\|\lambda_{0}\|_{L^{\infty}({\mathbb{R}})})\|u_{0}\|_{H^{2j+1}({\mathbb{R}})}+2^{\frac{p}{2}}\|u_{0}\|^{\frac{p}{2}}_{2}\|\partial_{x}u_{0}\|^{\frac{p+2}{2}}_{2}+\|\lambda\|_{\infty}\|u_{0}(\cdot,-\tau)\|_{2}.$

Using the Gagliardo-Nirenberg inequality, we have

(2.29) $\|\partial_{x}u_{0}\|_{2}\leq
C\|\partial^{2j+1}_{x}u_{0}\|^{\frac{1}{2j+1}}_{2}\|u_{0}\|_{2}^{1-\frac{1}{2j+1}}.$

Applying (2.29), we have

$\begin{split}\|v_{0}\|_{2}\leq&(1+\|\lambda_{0}\|_{L^{\infty}({\mathbb{R}})})\|u_{0}\|_{H^{2j+1}({\mathbb{R}})}\\\
&+2^{\frac{p}{2}}C\|\partial^{2j+1}_{x}u_{0}\|^{\frac{p+2}{2(2j+1)}}_{2}\|u_{0}\|_{2}^{\frac{p+2}{2}\left(1-\frac{1}{2j+1}\right)+\frac{p}{2}}+\|\lambda\|_{\infty}\|u_{0}(\cdot,-\tau)\|_{2}.\end{split}$

Then, Young inequality guarantees that

$\begin{split}\|v_{0}\|_{2}\leq&(1+\|\lambda_{0}\|_{L^{\infty}({\mathbb{R}})})\|u_{0}\|_{H^{2j+1}({\mathbb{R}})}+2^{\frac{p(2j+1)}{4j-p}}\|u_{0}\|^{\left(\frac{p+2}{2}\left(1-\frac{1}{2j+1}\right)+\frac{p-2}{2}\right)\frac{2(2j+1)}{4j-p}}_{2}\|u_{0}\|_{2}\\\
&+\|\partial_{x}^{2j+1}u_{0}\|_{2}+\|\lambda\|_{\infty}\|u_{0}(\cdot,-\tau)\|_{2}\end{split}$

and, consequently, leader to

(2.30) $\displaystyle\|v_{0}\|_{2}$ $\displaystyle\leq
C(\|u_{0}\|_{2})\|u_{0}\|_{H^{2j+1}({\mathbb{R}})}+\|\lambda\|_{\infty}\|u_{0}(\cdot,-\tau)\|_{2}$
$\displaystyle\leq
C(\|u_{0}\|_{2})\|u_{0}\|_{H^{2j+1}({\mathbb{R}})}+\|\lambda\|_{\infty}\|u_{0}\|_{C([-\tau,0];L^{2}(\mathbb{R})})$
$\displaystyle\leq(C(\|u_{0}\|_{2})+\|\lambda\|_{\infty})\|u_{0}\|_{C([-\tau,0];H^{2j+1}({\mathbb{R}}))}$

where
$C(s)=2+\|\lambda_{0}\|_{L^{\infty}({\mathbb{R}})}+2^{\frac{p(2j+1)}{4j-p}}s^{\left(\frac{p+2}{2}\left(1-\frac{1}{2j+1}\right)+\frac{p-2}{2}\right)\frac{2(2j+1)}{4j-p}}$.
Thanks to Proposition 2.10, we see that $v\in B_{0,T}$ and

$\|v\|_{{\mathcal{B}}_{T}}\leq\sigma(\|u\|_{{\mathcal{B}}_{T}})\|v_{0}\|_{C([-\tau,0];L^{2}({\mathbb{R}}))}.$

Combining (2.28) and (2.30), we get

(2.31)
$\|v\|_{{\mathcal{B}}_{T}}\leq\sigma(\beta_{0}(\|u_{0}\|_{2})\|u_{0}\|_{2})\left(C(\|u_{0}\|_{2})+\|\lambda\|_{\infty}\|\right)\|u_{0}\|_{C([-\tau,0];H^{2j+1}({\mathbb{R}}))}.$

which means

(2.32) $u,u_{t}\in L^{2}(0,T;H^{j}({\mathbb{R}})).$

Therefore,

(2.33) $u\in C([0,T];H^{j}({\mathbb{R}}))\hookrightarrow
C([0,T];C({\mathbb{R}})).$

On the other hand, note that $u^{p}u_{x},\lambda_{0}u$ and $\lambda
u(\cdot,t-\tau)$ belong to $L^{2}(0,T;L^{2}({\mathbb{R}}))$. Moreover,
$(-1)^{j+1}\partial_{x}^{2j+1}u+(-1)^{m}\partial_{x}^{2m}u=-u_{t}-u^{p}u_{x}-\lambda_{0}u-\lambda
u(\cdot,t-\tau)$ in $D^{\prime}(0,T,{\mathbb{R}})$. Hence,

$(-1)^{j+1}\partial_{x}^{2j+1}u+(-1)^{m}\partial_{x}^{2m}u=f\quad\in\quad
L^{2}(0,T;L^{2}({\mathbb{R}})).$

Taking Fourier transform, we have

(2.34)
$\widehat{u}=\frac{\widehat{f}+\widehat{u}}{[\left(1-((-1)^{j+1}(i\xi)^{2j+1})-(-1)^{j}(i\xi)^{2j}\right)]}$

and,

(2.35) $\displaystyle\|u(t)\|^{2}_{H^{2j+1}({\mathbb{R}})}\leq
C\left\\{\|f(t)\|_{2}^{2}+\|u(t)\|_{2}^{2}\right\\}$

where
$C=\sup_{\xi\in{\mathbb{R}}}\frac{1+|\xi|+|\xi|^{2}+|\xi|^{3}+\cdots+|\xi|^{2j+1}}{\left|1-((-1)^{j+1}(i\xi)^{2j+1}){\color[rgb]{0,0,0}-}(-1)^{j}(i\xi)^{2j}\right|}$.
Integrating (2.35) over $[0,T]$, we deduce that

(2.36) $u\in L^{2}(0,T;H^{2j+1}({\mathbb{R}})),$

achieving step 1.

Step 2: $u\in C(0,T;H^{2j+1}(\mathbb{R})$

Observe that, according to (2.32), $u_{t}\in
L^{2}(0,T;H^{-(2j+1)}({\mathbb{R}}))$. Then, considering the Hilbert triple
$H^{2j+1}({\mathbb{R}})\hookrightarrow H^{2m}({\mathbb{R}})\hookrightarrow
H^{-(2j+1)}({\mathbb{R}})$, by [34, Chapter III - Lemma 1.2], we have

$u\in C([0,T];H^{2m}({\mathbb{R}})).$

This implies further

(2.37) $\partial_{x}^{2m}u,\lambda_{0}u,\lambda u(\cdot,\cdot-\tau)\in
C([0,T];L^{2}({\mathbb{R}}))\cap L^{2}(0,T;H^{1}({\mathbb{R}})).$

On the other hand, note that

$\displaystyle\|u(t)^{p}u_{x}(t)-u(t_{0})^{p}u_{x}(t_{0})\|_{2}$
$\displaystyle\leq\|[u(t)^{p}-u(t_{0})^{p}]u_{x}(t)\|_{2}+\|u(t_{0})^{p}[u_{x}(t)-u_{x}(t_{0})]\|_{2}$
$\displaystyle\leq
C\left\\{\|(1+|u(t)|^{p-1}+|u(t_{0})|^{p-1})|u(t)-u(t_{0})|u_{x}(t)\|_{2}\right.$
$\displaystyle\left.+\|(1+|u(t_{0})|^{p})|u_{x}(t)-u_{x}(t_{0})|\|_{2}\right\\}$
$\displaystyle\leq
C\left\\{(1+\|u(t)\|^{p-1}_{\infty}+\|u(t_{0})\|^{p-1}_{\infty})\|u(t)-u(t_{0})\|_{\infty}\|u_{x}(t)\|_{2}\right.$
$\displaystyle\left.+(1+\|u(t_{0})\|^{p}_{\infty})\|u_{x}(t)-u_{x}(t_{0})\|_{2}\right\\}.$

Then, the regularity given in (2.33) ensures that

$\lim_{t\rightarrow t_{0}}\|u(t)^{p}u_{x}(t)-u(t_{0})^{p}u_{x}(t_{0})\|_{2}=0$

and, therefore $u^{p}u_{x}\in C([0,T];L^{2}({\mathbb{R}}))$. The results above
also guarantee that

(2.38) $u^{p}u_{x}\in C([0,T];L^{2}({\mathbb{R}}))\cap
L^{2}(0,T;H^{1}({\mathbb{R}})).$

Indeed, note that $(u^{p}u_{x})_{x}=pu^{p-1}u_{x}^{2}+u^{p}u_{xx}$, so it is
sufficient to combine (2.33), (2.36) and the following two estimates

$\displaystyle\|pu^{p-1}u_{x}^{2}\|_{L^{2}(0,T;L^{2}({\mathbb{R}}))}\leq
C\left\\{(1+\|u\|_{C([0,T];C({\mathbb{R}}))}^{p-1})\|u_{x}\|_{C([0,T];C({\mathbb{R}}))}\|u_{x}\|_{L^{2}([0,T];L^{2}({\mathbb{R}}))}\right\\}$

and

$\displaystyle\|u^{p}u_{xx}\|_{L^{2}(0,T;L^{2}({\mathbb{R}}))}$
$\displaystyle\leq
C\left\\{(1+\|u\|_{C([0,T];C({\mathbb{R}}))}^{p})\|u_{xx}\|_{L^{2}([0,T];L^{2}({\mathbb{R}}))}\right\\}.$

Since

(2.39) $\displaystyle(-1)^{j+1}\partial_{x}^{2j+1}u$
$\displaystyle=-(-1)^{m}\partial_{x}^{2m}u-u_{t}-(u^{p}u_{x})-\lambda_{0}u-\lambda,u(\cdot,t-\tau),$

from (2.32), (2.37), (2.38), we obtain

(2.40) $u\in L^{2}(0,T;H^{3j+1}({\mathbb{R}}))\hookrightarrow
L^{2}(0,T;H^{2j+2}({\mathbb{R}})).$

Finally, considering the Hilbert triple $H^{2j+2}({\mathbb{R}})\hookrightarrow
H^{2j+1}({\mathbb{R}})\hookrightarrow H^{-(2j+2)}({\mathbb{R}})$, [34, Chapter
III - Lemma 1.2] gives

(2.41) $u\in C([0,T];H^{2j+1}({\mathbb{R}})),$

and step 2 is proved. Note that (2.40) and (2.41) imply that
$u\in{\mathcal{B}}_{2j+1,T}$.

Step 3: The following estimate holds

$\|u\|_{C([0,T];H^{2j+1}({\mathbb{R}}))}\leq\sigma_{1}(\|u_{0}\|_{2})\|u_{0}\|_{C([-\tau,0];H^{2j+1}({\mathbb{R}}))}.$

Indeed, according to (2.35), we get

(2.42) $\displaystyle\|u(t)\|_{H^{2j+1}({\mathbb{R}})}\leq
C\left\\{\|u_{t}(t)\|_{2}+\|u^{p}(t)u_{x}(t)\|_{2}+\|\lambda_{0}u(t)\|_{2}+\|\lambda
u(t-\tau)\|_{2}+\|u(t)\|_{2}\right\\}.$

Now, by using Gagliardo-Nirenberg inequality, we obtain

$\|u(t)^{p}u_{x}(t)\|_{2}\leq
2^{\frac{1}{p}}\|u(t)\|_{2}^{\frac{p}{2}}\|u_{x}(t)\|_{2}^{\frac{p+2}{2}}\leq
2^{\frac{1}{p}}C\|u(t)\|_{2}^{\frac{4j(p+1)+p}{2(2j+1)}}\|u(t)\|_{H^{2j+1}({\mathbb{R}})}^{\frac{p+2}{2(2j+1)}}.$

Moreover, Young inequality gives

$\displaystyle\|u(t)^{p}u_{x}(t)\|_{2}$ $\displaystyle\leq
C\|u(t)\|_{2}^{\frac{4j(p+1)+p}{4j-p}}+\frac{1}{2C}\|u(t)\|_{H^{2j+1}({\mathbb{R}})}$

Replacing the estimate above into the inequality (2.42), yields that

$\displaystyle\|u(t)\|_{H^{2j1}({\mathbb{R}}))}\leq
C\left\\{\|u_{t}\|_{{\mathcal{B}}_{T}}+(1+\|\lambda_{0}\|_{\infty})\|u\|_{{\mathcal{B}}_{T}}+\|u\|_{{\mathcal{B}}_{T}}^{\frac{4j(p+1)+p}{4j-p}}+\|\lambda\|_{\infty}\|u(t-\tau)\|_{2}\right\\}.$

Then, using (2.28) and (2.31) it follows that

$\displaystyle\|u(t)\|_{H^{2j+1}({\mathbb{R}}))}\leq$ $\displaystyle
C\left\\{\sigma(\beta_{0}(\|u_{0}\|_{2})\|u_{0}\|_{2})C(\|u_{0}\|_{2})\|u_{0}\|_{H^{2j+1}({\mathbb{R}})}+(C+\|\lambda_{0}\|_{\infty})\beta_{0}(\|u_{0}\|_{2})\|u_{0}\|_{2}\right.$
$\displaystyle\left.+\beta_{0}^{\frac{4j(p+1)+p}{4j-p}}(\|u_{0}\|_{2})\|u_{0}\|_{2}^{\frac{4j(p+1)+p}{4j-p}-1}\|u_{0}\|_{2}+\|\lambda\|_{\infty}\|u(t-\tau)\|_{2}\right\\}$
$\displaystyle\leq$
$\displaystyle\bar{\sigma}_{1}(\|u_{0}\|_{2})\|u_{0}\|_{C([-\tau,0];H^{2j+1}({\mathbb{R}}))}+\|\lambda\|_{\infty}\|u(t-\tau)\|_{2},$

where

$\bar{\sigma}_{1}(s)=C\left\\{\sigma(\beta_{0}(s)s)C(s)+(C+\|b\|_{\infty})\beta_{0}(s)+\beta_{0}^{\frac{4j(p+1)+p}{4j-p}}(s)s^{\frac{4j(p+1)+p}{4j-p}-1}\right\\}.$

Thus, we obtain that

$\displaystyle\|u\|_{C([0,T];H^{2j+1}({\mathbb{R}})))}\leq$
$\displaystyle\bar{\sigma}_{1}(\|u_{0}\|_{2})\|u_{0}\|_{C([-\tau,0];H^{2j+1}({\mathbb{R}}))}+\|\lambda\|_{\infty}\|u\|_{C([-\tau,T-\tau];L^{2}({\mathbb{R}}))}$
$\displaystyle\leq$
$\displaystyle\bar{\sigma}_{1}(\|u_{0}\|_{2})\|u_{0}\|_{C([-\tau,0];H^{2j+1}({\mathbb{R}}))}+\|\lambda\|_{\infty}\|u_{0}\|_{C([-\tau,0];L^{2}({\mathbb{R}}))}$
$\displaystyle+\|\lambda\|_{\infty}\|u\|_{C([0,T-\tau];L^{2}({\mathbb{R}}))}$
$\displaystyle\leq$
$\displaystyle\left(\bar{\sigma}_{1}(\|u_{0}\|_{2})+\|\lambda\|_{\infty}\right)\|u_{0}\|_{C([-\tau,0];H^{2j+1}({\mathbb{R}}))}+\|\lambda\|_{\infty}\|u\|_{{\mathcal{B}}_{T}}$

Finally, from Theorem 2.9 and the previous inequality, we deduce that

(2.43) $\displaystyle\|u\|_{C([0,T];H^{2j+1}({\mathbb{R}})))}$
$\displaystyle\leq\sigma_{1}(\|u_{0}\|_{2})\|u_{0}\|_{C([-\tau,0];H^{2j+1}({\mathbb{R}}))},$

where
$\sigma_{1}(s)=\bar{\sigma}_{1}(s)+\|\lambda\|_{\infty}(\beta_{0}(s)+1)$,
finishing the step 3.

Step 4: The following estimate is verified

$\|\partial_{x}^{3j+1}u\|_{L^{2}(0,T;L^{2}({\mathbb{R}}))}\leq\sigma_{3}(\|u_{0}\|_{2})\|u_{0}\|_{C([-\tau,0];H^{2j+1}({\mathbb{R}}))}.$

In fact, from the equation (2.39), we deduce that

$\displaystyle\int_{0}^{T}\|u(t)\|_{H^{3j+1}}^{2}dt$
$\displaystyle=\int_{0}^{T}\|\partial_{x}^{3j+1}u(t)\|_{2}^{2}dt+\int_{0}^{T}\|u(t)\|_{2}^{2}dt$
$\displaystyle\leq\int_{0}^{T}\|u(t)\|_{2}^{2}dt+\int_{0}^{T}\|\partial_{x}^{2m+j}u(t)\|_{2}^{2}dt+\int_{0}^{T}\|\partial_{x}^{j}u_{t}(t)\|_{2}^{2}dt$
$\displaystyle\quad+\int_{0}^{T}\|\partial_{x}^{j}(u^{p}u_{x})\|_{2}^{2}dt+\int_{0}^{T}\|\partial_{x}^{j}(\lambda_{0}u(t))\|_{2}^{2}dt+\int_{0}^{T}\|\partial_{x}^{j}(\lambda_{0}u(t-\tau))\|_{2}^{2}dt$
$\displaystyle\leq
CT\left(\|u\|_{C([0,T];H^{2j+1})}+\|u\|_{C([0,T];H^{2m+j})}\right)+\|u_{t}\|_{B_{0,T}}$
$\displaystyle\quad+\int_{0}^{T}\|\partial_{x}^{j}(u^{p}u_{x})\|_{2}^{2}dt+\int_{0}^{T}\|\partial_{x}^{j}(\lambda_{0}u(t))\|_{2}^{2}dt+\int_{0}^{T}\|\partial_{x}^{j}(\lambda_{0}u(t-\tau))\|_{2}^{2}dt.$

Note that $H^{2m+j}({\mathbb{R}})\subset H^{3j+1}({\mathbb{R}})$, thus using
step 4 and Proposition 2.10, there exists a function $\sigma_{2}$ such that

(2.44)
$\begin{split}\int_{0}^{T}\|u(t)\|_{H^{3j+1}}^{2}dt&\leq\sigma_{2}\left(\|u_{0}\|_{2}\right)\|u_{0}\|_{C([-\tau,0];H^{2j+1})}\\\
&+\int_{0}^{T}\|\partial_{x}^{j}(u^{p}u_{x})\|_{2}^{2}dt+\int_{0}^{T}\|\partial_{x}^{j}(\lambda_{0}u(t))\|_{2}^{2}dt+\int_{0}^{T}\|\partial_{x}^{j}(\lambda_{0}u(t-\tau))\|_{2}^{2}dt.\end{split}$

Now, we have to estimate the last three integrals in the RHS of (2.44). For
the second and third terms, observe that

$\partial_{x}^{j}(\lambda_{0}u)=\sum_{k=0}^{j}\left(\begin{array}[]{c}j\\\
k\end{array}\right)\partial_{x}^{k}(\lambda_{0})\partial_{x}^{j-k}(u)$

and

$\partial_{x}^{j}(\lambda
u(\cdot,t-\tau))=\sum_{k=0}^{j}\left(\begin{array}[]{c}j\\\
k\end{array}\right)\partial_{x}^{k}(\lambda)\partial_{x}^{j-k}(u(\cdot,t-\tau)),$

thus we have that

$\begin{split}\int_{0}^{T}\|\partial_{x}^{j}(\lambda_{0}u)(t)\|_{2}^{2}dt\leq&\sum_{k=0}^{j}\int_{0}^{T}\left(\begin{array}[]{c}j\\\
k\end{array}\right)\|\partial_{x}^{k}(\lambda_{0})\partial_{x}^{j-k}(u)(t)\|_{2}^{2}dt\\\
=&\int_{0}^{T}\|\lambda_{0}\|_{\infty}^{2}\|\partial_{x}^{j}(u)(t)\|_{2}^{2}dt\\\
&+\sum_{k=1}^{j}\int_{0}^{T}\left(\begin{array}[]{c}j\\\
k\end{array}\right)\|\partial_{x}^{k}(\lambda_{0})\|_{2}^{2}\|\partial_{x}^{j-k}(u)(t)\|_{\infty}^{2}dt\\\
\leq&C\|\lambda_{0}\|_{H^{1}({\mathbb{R}})}^{2}\int_{0}^{T}\|\partial_{x}^{j}(u)(t)\|_{2}^{2}dt\\\
&+\sum_{k=1}^{j}\int_{0}^{T}\left(\begin{array}[]{c}j\\\
k\end{array}\right)\|\lambda_{0}\|_{H^{k}({\mathbb{R}})}^{2}\|u(t)\|_{H^{1+j-k}({\mathbb{R}})}^{2}dt\\\
\leq&CT\|\lambda_{0}\|_{H^{j}({\mathbb{R}})}^{2}\|u\|_{C([0,T];H^{2j+1}({\mathbb{R}})}^{2}.\end{split}$

Similarly, it follows that

$\begin{split}\int_{0}^{T}\|\partial_{x}^{j}(\lambda_{0}u)(t)\|_{2}^{2}dt\leq&C\|\lambda_{0}\|_{H^{1}({\mathbb{R}})}^{2}\int_{0}^{T}\|\partial_{x}^{j}(u)(t-\tau)\|_{2}^{2}dt\\\
&+\sum_{k=1}^{j}\int_{0}^{T}\left(\begin{array}[]{c}j\\\
k\end{array}\right)\|\lambda_{0}\|_{H^{k}({\mathbb{R}})}^{2}\|u(t-\tau)\|_{H^{1+j-k}({\mathbb{R}})}^{2}dt\\\
=&C\|\lambda_{0}\|_{H^{1}({\mathbb{R}})}^{2}\int_{-\tau}^{T-\tau}\|\partial_{x}^{j}(u)(t)\|_{2}^{2}dt\\\
&+\sum_{k=1}^{j}\int_{-\tau}^{T-\tau}\left(\begin{array}[]{c}j\\\
k\end{array}\right)\|\lambda_{0}\|_{H^{k}({\mathbb{R}})}^{2}\|u(t)\|_{H^{1+j-k}({\mathbb{R}})}^{2}dt\\\
\leq&CT\|\lambda_{0}\|_{H^{j}({\mathbb{R}})}^{2}\left(\|u\|_{C([0,T];H^{2j+1}({\mathbb{R}})}^{2}+\int_{-\tau}^{0}\|u_{0}(t)\|_{H^{j}({\mathbb{R}})}^{2}dt\right).\end{split}$

Now, we concentrate in estimate the nonlinear term, that is, the first term of
(2.44). Observing that

$\displaystyle\partial_{x}^{j}(u^{p}u_{x})=\sum_{k=0}^{j}\left(\begin{array}[]{c}j\\\
k\end{array}\right)\partial_{x}^{k}(u^{p})\partial_{x}^{j-k}(u_{x}),$

we get

$\displaystyle\int_{0}^{T}\|\partial_{x}^{j}(u^{p}u_{x})(t)\|_{2}^{2}dt$
$\displaystyle\leq\sum_{k=0}^{j}\left(\begin{array}[]{c}j\\\
k\end{array}\right)\int_{0}^{T}\|\partial_{x}^{k}(u^{p})(t)\|_{\infty}^{2}\|\partial_{x}^{j-k}(u_{x})(t)\|_{2}^{2}dt$
$\displaystyle=\sum_{k=0}^{j}\left(\begin{array}[]{c}j\\\
k\end{array}\right)\int_{0}^{T}\|\partial_{x}^{k}(u^{p})(t)\|_{\infty}^{2}\|u(t)\|_{H^{1+j-k}({\mathbb{R}})}^{2}dt$
$\displaystyle\leq C\sum_{k=0}^{j}\left(\begin{array}[]{c}j\\\
k\end{array}\right)\int_{0}^{T}\|\partial_{x}^{k}(u^{p})(t)\|_{\infty}^{2}\|u(t)\|_{H^{2j+1}({\mathbb{R}})}^{2}dt.$

On the other hand, note that the $k$-order derivative of the function $u^{p}$
can be written in the following way

$\displaystyle\partial_{x}^{k}(u^{p})=C_{0,p}u^{p-1}\partial_{x}^{k}u+\sum_{n=1}^{k-1}C_{n,p}u^{p-n}F_{n}(u)+C_{k,p}u^{p-k}u_{x}^{k},$

where $C_{n,p}$ is a constant given by

$C_{n,p}=M_{n}\prod_{i=0}^{n}(p-i)$

with $M_{n}\in{\mathbb{N}}$ and $F_{n}(u)$ is a differential operator
involving sums and products of derivatives of $u$ with order less than $n+1$.
Thanks to this fact, we have the following estimate:

$\displaystyle\|\partial_{x}^{k}(u^{p})\|_{\infty}\leq$
$\displaystyle|C_{0,p}|\|u\|_{\infty}^{p-1}\|\partial_{x}^{k}u\|_{\infty}+\sum_{n=1}^{k-1}|C_{n,p}|\|u\|_{\infty}^{p-n}\|F_{n}(u)\|_{\infty}+|C_{k,p}|\|u\|_{\infty}^{p-k}\|u_{x}\|_{\infty}^{k}$
$\displaystyle\leq$ $\displaystyle
C\left(|C_{0,p}|\|u\|_{H^{2j+1}({\mathbb{R}})}^{p-1}\|u\|_{H^{k+1}({\mathbb{R}})}+\sum_{n=1}^{k-1}|C_{n,p}|\|u\|_{H^{2j+1}({\mathbb{R}})}^{p-n}\|u\|_{H^{n+2}({\mathbb{R}})}^{m(k)}\right.$
$\displaystyle\left.+|C_{k,p}|\|u\|_{H^{2j+1}({\mathbb{R}})}^{p-k}\|u\|_{H^{2}({\mathbb{R}})}^{k}\right),$

where $m(n)\in{\mathbb{N}}$ and $C>0$ is a constant. Since
$H^{2j+1}({\mathbb{R}})\subset H^{k+1}({\mathbb{R}})$,
$H^{2j+1}({\mathbb{R}})\subset H^{n+1}({\mathbb{R}})$ and
$H^{2j+1}({\mathbb{R}})\subset H^{2}({\mathbb{R}})$, it follows that

$\displaystyle\|\partial_{x}^{k}(u^{p})\|_{\infty}$ $\displaystyle\leq
C\left(|C_{0,p}|\|u\|_{H^{2j+1}({\mathbb{R}})}^{p}+\sum_{n=1}^{k-1}|C_{n,p}|\|u\|_{H^{2j+1}({\mathbb{R}})}^{p-n+m(k)}+|C_{k,p}|\|u\|_{H^{2j+1}({\mathbb{R}})}^{p}\right).$

Hence, it follows that

$\begin{split}\int_{0}^{T}\|\partial_{x}^{j}(u^{p}u_{x})(t)\|_{2}^{2}dt\leq&C|C_{0,p}|^{2}\int_{0}^{T}\|u(t)\|_{H^{2j+1}({\mathbb{R}})}^{2(p+1)}dt\\\
&+C\sum_{k=0}^{j}\sum_{n=0}^{k}\left(\begin{array}[]{c}j\\\
k\end{array}\right)|C_{n,p}|^{2}\int_{0}^{T}\|u(t)\|_{H^{2j+1}({\mathbb{R}})}^{2(p-n+m(k)+1)}dt\\\
&+C\sum_{k=0}^{j}\left(\begin{array}[]{c}j\\\
k\end{array}\right)|C_{k,p}|^{2}\int_{0}^{T}\|u(t)\|_{H^{2j+1}({\mathbb{R}})}^{2(p+1)}dt.\end{split}$

Finally, we obtain

(2.45)
$\begin{split}\int_{0}^{T}\|\partial_{x}^{j}(u^{p}u_{x})(t)\|_{2}^{2}dt\leq&CT\left(|C_{0,p}|^{2}\|u\|_{C([0,T];H^{2j+1}({\mathbb{R}}))}^{2(p+1)}\right.\\\
&+\sum_{k=0}^{j}\sum_{n=0}^{k}\left(\begin{array}[]{c}j\\\
k\end{array}\right)|C_{n,p}|^{2}\|u(t)\|_{C([0,T];H^{2j+1}({\mathbb{R}}))}^{2(p-n+m(k)+1)}\\\
&\left.+\sum_{k=0}^{j}\left(\begin{array}[]{c}j\\\
k\end{array}\right)|C_{k,p}|^{2}\|u(t)\|_{C([0,T];H^{2j+1}({\mathbb{R}}))}^{2(p+1)}\right).\end{split}$

Therefore, using estimates (2.44), (2.4.2), (2.4.2), (2.45) and step 3, step 4
holds, and the theorem is verified. ∎

###  2.5. Interpolation arguments

In this part of the work, we present the proof of the well-posedness result
for the system (1.2). To that, let us introduce an interpolation argument due
to Tartar [33] and adapted by Bona and Scott [8, Theorem 4.3].

Let $B_{0}$ and $B_{1}$ be two Banach spaces, where $B_{1}\subset B_{0}$ with
the inclusion map continuous. Let $f\in B_{0}$ and define

$\displaystyle K(f,t)=\inf_{g\in
B_{1}}\left\\{\|f-g\|_{B_{0}}+t\|g\|_{B_{1}}\right\\},$

for $t\geq 0$. For $0<\theta<1$ and $1\leq p\leq+\infty$, consider the set

$\displaystyle{\mathcal{B}}_{\theta,p}:=[B_{0},B_{1}]_{\theta,p}=\left\\{f\in
B_{0}:\|f\|_{\theta,p}:\left(\int_{0}^{\infty}K(f,t)t^{-\theta
p-1}dt\right)^{\frac{1}{p}}<\infty\right\\},$

with the usual modification for the case $p=\infty$. Then, $B_{\theta,p}$ is a
Banach space with norm $\|\cdot\|_{\theta,p}$. Given two pairs
$(\theta_{1},p_{1})$ and $(\theta_{2},p_{2})$ as above,
$(\theta_{1},p_{1})\prec(\theta_{2},p_{2})$ will denote

$\left\\{\begin{tabular}[]{l l}$\theta_{1}<\theta_{2}$&or\\\
$\theta_{1}=\theta_{3}$&and $p_{1}>p_{2}$.\end{tabular}\right.$

If $(\theta_{1},p_{1})\prec(\theta_{2},p_{2})$, then
${\mathcal{B}}_{\theta_{2},p_{2}}\subset{\mathcal{B}}_{\theta_{1},p_{1}}$ with
the inclusion map continuous. Then, the following result, proved by [8,
Theorem 4.3], holds.

######  Theorem 2.12.

Let $B_{0}^{j}$ and $B_{1}^{j}$ be Banach spaces such that $B_{1}^{j}\subset
B_{0}^{j}$ with continuous inclusion mappings, for $j=1,2$. Let $\alpha$ and
$q$ lie in the ranges $0<\alpha<1$ and $1\leq q\leq\infty$. Suppose that
$\mathcal{A}$ is a mapping satisfying:

  1. (i)

$\mathcal{A}:{\mathcal{B}}_{\alpha,q}^{1}\rightarrow B_{0}^{2}$ and, for
$f,g\in{\mathcal{B}}_{\alpha,q}^{1}$,

$\|\mathcal{A}f-\mathcal{A}g\|_{B_{0}^{2}}\leq
C_{0}\left(\|f\|_{{\mathcal{B}}_{\alpha,q}^{1}}+\|g\|_{{\mathcal{B}}_{\alpha,q}^{1}}\right)\|f-g\|_{B_{0}^{1}}.$

  2. (ii)

$\mathcal{A}:B_{1}^{1}\rightarrow B_{1}^{2}$ and, for
$h\in{\mathcal{B}}_{1}^{1}$,

$\|\mathcal{A}h\|_{B_{1}^{2}}\leq
C_{1}\left(\|h\|_{{\mathcal{B}}_{\alpha,q}^{1}}\right)\|h\|_{B_{1}^{1}},$

where $C_{j}:{\mathbb{R}}^{+}\rightarrow{\mathbb{R}}^{+}$ are continuous
nondecreasing functions, for $j=0,1$. Then, if $(\theta,p)\geq(\alpha,q)$,
$\mathcal{A}$ maps ${\mathcal{B}}_{\theta,p}^{1}$ into
${\mathcal{B}}_{\theta,p}^{2}$ and, for $f\in{\mathcal{B}}_{\theta,p}^{1}$, we
have

$\|\mathcal{A}f\|_{{\mathcal{B}}_{\theta,p}^{2}}\leq
C\left(\|f\|_{{\mathcal{B}}_{\alpha,q}^{1}}\right)\|f\|_{{\mathcal{B}}_{\theta,p}^{1}},$

where $C(r)=4C_{0}(4r)^{1-\theta}C_{1}(3r)^{\theta}$, with $r>0$.

It follows from Theorem 2.9 that, for each fixed $T>0$, the solution map

(2.46)
$\mathcal{A}:C([-\tau,0];L^{2}({\mathbb{R}}))\rightarrow{\mathcal{B}}_{0,T},\quad\mathcal{A}u_{0}=u$

is well-defined. Moreover, we have the following result:

######  Proposition 2.13.

The solution map (2.46) is locally Lipschitz continuous, that is, there exists
a continuous function
$C_{0}:{\mathbb{R}}^{+}\times(0,\infty)\rightarrow{\mathbb{R}}^{+}$,
nondecreasing in its first variable, such that, for all $u_{0},v_{0}\in
C([-\tau,0];L^{2}({\mathbb{R}}))$, we have

$\|\mathcal{A}u_{0}-\mathcal{A}v_{0}\|_{0,T}\leq
C_{0}\left(\|u_{0}\|_{2}+\|v_{0}\|_{2},T\right)\|u_{0}-v_{0}\|_{C([-\tau,0];L^{2}({\mathbb{R}}))}.$

###### Proof.

Let $0<\theta\leq T$ and $n=\left[\frac{T}{\theta}\right]$. Theorem 2.9
ensures that

(2.47)
$\|\mathcal{A}u_{0}\|_{{\mathcal{B}}_{0,\theta}}\leq\beta_{0}(\|u_{0}\|_{2})\|u_{0}\|_{C([-\tau,0];L^{2}(\mathbb{R}))},$

where $\beta_{0}(s)$ is a constant function given by
$\beta_{0}=C_{T}\left(1+\left(\|\lambda\|_{\infty}\tau^{1/2}+\|\lambda\|_{\infty}^{1/2}\right)\right)$
and

$\|\mathcal{A}u_{0}-\mathcal{A}v_{0}\|_{0,\theta}\leq
C_{\theta}\left\\{\|u_{0}-v_{0}\|_{2}+\|M(\mathcal{A}u_{0})-M(\mathcal{A}v_{0})\|_{L^{1}(0,\theta;L^{2}({\mathbb{R}}))}\right\\},$

where $C_{\theta}=2e^{\theta\|b\|_{\infty}}$. Moreover, thanks to the Lemma
2.6 we have

$\begin{split}\|\mathcal{A}u_{0}-\mathcal{A}v_{0}\|_{{\mathcal{B}}_{0,\theta}}\leq&C_{\theta}\|u_{0}-v_{0}\|_{2}+C_{\theta}C\left\\{2^{\frac{p}{2}}\theta^{\frac{2j-p}{4j}}\left(\|\mathcal{A}u_{0}\|_{{\mathcal{B}}_{0,\theta}}^{p}+\|\mathcal{A}u_{0}\|_{{\mathcal{B}}_{0,\theta}}\|\mathcal{A}v_{0}\|_{{\mathcal{B}}_{0,\theta}}^{p-1}\right.\right.\\\
&\left.\left.+\|\mathcal{A}v_{0}\|_{{\mathcal{B}}_{0,\theta}}^{p}\right)+2^{\frac{1}{2}}\theta^{\frac{1}{4}}\|\mathcal{A}u_{0}\|_{{\mathcal{B}}_{0,\theta}}\right\\}\|\mathcal{A}u_{0}-\mathcal{A}v_{0}\|_{\mathcal{B}_{0,\theta}}.\end{split}$

Now, inequality (2.47) together with the following estimate

$\begin{split}\|\mathcal{A}u_{0}-\mathcal{A}v_{0}\|_{0,\theta}\leq&C_{\theta}\|u_{0}-v_{0}\|_{2}+C_{\theta}C\left\\{2^{\frac{p}{2}}\theta^{\frac{2j-p}{4j}}\beta_{0}^{p}\left(\|u_{0}\|_{C([-\tau,0];L^{2}(\mathbb{R}))}^{p}\right.\right.\\\
&\left.\left.+\|\ u_{0}\|_{C([-\tau,0];L^{2}(\mathbb{R}))}\|\
v_{0}\|_{C([-\tau,0];L^{2}(\mathbb{R}))}^{p-1}+\|\
v_{0}\|_{C([-\tau,0];L^{2}(\mathbb{R}))}^{p}\right)\right.\\\
&\left.+2^{\frac{1}{2}}\beta_{0}\theta^{\frac{1}{4}}\|u_{0}\|_{C([-\tau,0];L^{2}(\mathbb{R}))}\right\\}\|\mathcal{A}u_{0}-\mathcal{A}v_{0}\|_{\mathcal{B}_{0,\theta}},\end{split}$

yields that

(2.48) $\|\mathcal{A}u_{0}-\mathcal{A}v_{0}\|_{{\mathcal{B}}_{0,\theta}}\leq
2C_{T}\|u_{0}-v_{0}\|_{2},$

choosing $\theta$ small enough. Observe that denoting

$B_{0,[k\theta,(k+1)\theta]}:=C\left([k\theta,(k+1)\theta];L^{2}({\mathbb{R}})\right)\cap
L^{2}(k\theta,(k+1)\theta;H^{j}({\mathbb{R}})).$

with norm $\|\cdot\|_{{\mathcal{B}}_{0,[k\theta,(k+1)\theta]}}$, in a
analogously way as done to (2.48), we can deduce thanks to estimate (2.47)
that

$\begin{split}\|\mathcal{A}u_{0}-\mathcal{A}v_{0}\|_{{\mathcal{B}}_{0,[k\theta,(k+1)\theta]}}\leq&C_{\theta}\|u(k\theta)-v(k\theta)\|_{2}+C_{\theta}C\beta_{0}\left\\{2^{\frac{p}{2}}\theta^{\frac{2j-p}{4j}}\beta_{0}^{p}\left(\|u_{0}\|_{C([-\tau,0];L^{2}(\mathbb{R}))}^{p}\right.\right.\\\
&\left.\left.+\|\ u_{0}\|_{C([-\tau,0];L^{2}(\mathbb{R}))}\|\
v_{0}\|_{C([-\tau,0];L^{2}(\mathbb{R}))}^{p-1}+\|\
v_{0}\|_{C([-\tau,0];L^{2}(\mathbb{R}))}^{p}\right)\right.\\\
&\left.+2^{\frac{1}{2}}\beta_{0}\theta^{\frac{1}{4}}\|u_{0}\|_{C([-\tau,0];L^{2}(\mathbb{R}))}\right\\}\|\mathcal{A}u_{0}-\mathcal{A}v_{0}\|_{{\mathcal{B}}_{0,[k\theta,(k+1)\theta]}}.\end{split}$

Lastly, we get

(2.49)
$\displaystyle\|\mathcal{A}u_{0}-\mathcal{A}v_{0}\|_{{\mathcal{B}}_{0,[k\theta,(k+1)\theta]}}\leq
2C_{T}\|u(k\theta)-v(k\theta)\|_{2},\quad k=0,1,...,n-1.$

On the other hand, note that (2.48) and (2.49) imply that

$\displaystyle\|\mathcal{A}u_{0}-\mathcal{A}v_{0}\|_{{\mathcal{B}}_{0,[k\theta,(k+1)\theta]}}\leq
2^{k}C_{T}^{k}\|u_{0}-v_{0}\|_{2},\quad k=0,1,...,n-1,$

and, therefore,

$\|\mathcal{A}u_{0}-\mathcal{A}v_{0}\|_{{\mathcal{B}}_{0,[k\theta,(k+1)\theta]}}\leq
2^{n}C_{T}^{n}\|u_{0}-v_{0}\|_{2}.$

Finally,

$\displaystyle\|\mathcal{A}u_{0}-\mathcal{A}v_{0}\|_{{\mathcal{B}}_{0,T}}$
$\displaystyle\leq\sum_{k=0}^{n-1}\|\mathcal{A}u_{0}-\mathcal{A}v_{0}\|_{{\mathcal{B}}_{0,[k\theta,(k+1)\theta]}}\leq\sum_{k=0}^{n-1}2^{n}C_{T}^{n}\|u_{0}-v_{0}\|_{2}$
$\displaystyle\leq 2^{n}C_{T}^{n}n\|u_{0}-v_{0}\|_{2}\leq
C_{0}(\|u_{0}\|_{2}+\|v_{0}\|_{2})\|u_{0}-v_{0}\|_{C([-\tau,0];L^{2}(\mathbb{R}))},$

where $C_{0}(s)=\frac{T}{\theta(s)}\left[2C_{T}\right]^{\frac{T}{\theta(s)}}$,
getting the proposition. ∎

We are in a position to prove the main result of this section.

###### Proof of Theorem 1.1.

We define

$B_{0}^{1}=L^{2}({\mathbb{R}}),\quad B_{0}^{2}={\mathcal{B}}_{0,T},\quad
B_{1}^{1}=H^{2j+1}({\mathbb{R}})\quad\text{and}\quad
B_{1}^{2}={\mathcal{B}}_{2j+1,T}.$

Thus,

${\mathcal{B}}_{\frac{s}{2j+1},2}^{1}=[L^{2}({\mathbb{R}}),H^{2j+1}({\mathbb{R}})]_{\frac{s}{2j+1},2}=H^{s}({\mathbb{R}})\quad\text{and}\quad{\mathcal{B}}_{\frac{s}{2j+1},2}^{2}=[B_{0,T},B_{2j+1,T}]_{\frac{s}{2j+1},2}=B_{s,T}.$

Combining Proposition 2.13 and Theorem 2.11 we obtain $(i)$ and $(ii)$ in the
Theorem 2.12. Then, Theorem 2.12 gives the existence of the solution to the
equation (1.2), and Theorem 1.1 follows. ∎

##  3\. Exponential stabilization: Damping and delayed system

This section is devoted to proving the exponential stabilization for the
system (2.1) in $L^{2}({\mathbb{R}})$. In this way, first, we prove the result
for the linear system and, after that, we extend to the nonlinear one. To do
that consider the following Lyapunov functionals

(3.1) $E(t):=E(u(t))=\frac{1}{2}\int_{\mathbb{R}}u^{2}(x,t)dx$

and, for $\lambda\in L^{\infty}(\mathbb{R}),$

(3.2)
${\mathcal{E}}(t):={\mathcal{E}}(u(t))=\frac{1}{2}\int_{\mathbb{R}}u^{2}(x,t)dx+\frac{1}{2}\int_{t-\tau}^{t}\int_{\mathbb{R}}e^{-(t-s)}|\lambda(x)|u^{2}(x,s)\
dx\ ds.$

###  3.1. Asymptotic behavior: Linearized system with localized damping

The first result ensures the exponential stability for the linear system (2.1)
and can be read as follows.

######  Theorem 3.1.

Let $\lambda,\lambda_{0}\in L^{\infty}(\mathbb{R})$ and $\lambda_{0}$ be
constants satisfying (1.10). If there is a constant $\gamma>0$ and a function
$\beta\in L^{p}(\mathbb{R}),$ for $1\leq p<\infty$, such that the function
$\lambda$ verifies (1.12) and (1.13), then the problem (2.1) is exponentially
stable. Particularly, solutions $u$ of (2.1) satisfies the following
inequality

(3.3) ${\mathcal{E}}(t)\leq C(u_{0})e^{-\nu t},$

where $\nu$ and $C(u_{0})$ are defined by (1.16) and (1.17), respectively.

###### Proof.

Using the same argument as done in [12, Th. 4.7] we consider $u_{0}\in
H^{2j+1},$ then $u\in H^{2j+1}$. Taking the derivative in $t$ of
${\mathcal{E}}(t)$ we get

$\begin{split}\frac{d{\mathcal{E}}}{dt}(t)=&\int_{\mathbb{R}}u(t)({\color[rgb]{0,0,0}-}(-1)^{j}\partial^{2j}_{x}u(t)-\lambda_{0}u(t)-\lambda
u(t-\tau))dx+\frac{1}{2}\int_{\mathbb{R}}|\lambda|u^{2}(t)dx\\\
&-\frac{1}{2}e^{-\tau}\int_{\mathbb{R}}|\lambda|u^{2}(t-\tau)dx-\frac{1}{2}\int_{t-\tau}^{t}\int_{\mathbb{R}}e^{-(t-s)}|\lambda(x)|u^{2}(x,s)dx\,ds,\end{split}$

since we have

$\int_{\mathbb{R}}(-1)^{j+1}(\partial_{x}^{2j+1}u)udx=0.\quad\text{for
all}\quad u\in H^{2j+1}.$

So, integrating by parts, and using the Young inequality, remembering that
(1.10) and (1.12) are satisfied, we have

$\displaystyle\frac{d{\mathcal{E}}}{dt}(t)\leq$
$\displaystyle-\int_{\mathbb{R}}(\partial^{j}_{x}u)^{2}(t)dx-\gamma_{0}\int_{\mathbb{R}}u^{2}(t)dx+\frac{e^{\tau}+1}{2}\int_{\mathbb{R}}|\lambda(x)|u^{2}(t)dx$
$\displaystyle-\frac{1}{2}\int_{t-\tau}^{t}\int_{\mathbb{R}}e^{-(t-s)}|\lambda(x)|u^{2}(x,s)dxds$
$\displaystyle\leq$
$\displaystyle-\int_{\mathbb{R}}(\partial^{j}_{x}u)^{2}(t)dx-(\gamma_{0}-\gamma)\int_{\mathbb{R}}u^{2}(t)dx+\int_{\mathbb{R}}\beta(x)u^{2}(t)dx$
$\displaystyle-\frac{1}{2}\int_{t-\tau}^{t}\int_{\mathbb{R}}e^{-(t-s)}|\lambda(x)|u^{2}(x,s)\
dx\ ds,$

thanks to the fact that

$(-1)^{j}\int_{\mathbb{R}}u\partial^{2j}_{x}udx=\int_{\mathbb{R}}(\partial^{j}_{x}u)^{2}dx.$

The Hölder inequality ensures that

(3.4)
$\frac{d{\mathcal{E}}}{dt}(t)\leq-\|\partial^{j}_{x}u(t)\|_{2}^{2}-(\gamma_{0}-\gamma)\|u(t)\|_{2}^{2}+\|\beta\|_{q}\|u\|_{2q^{\prime}}^{2}-\frac{1}{2}\int_{t-\tau}^{t}\int_{\mathbb{R}}e^{-(t-s)}|\lambda(x)|u^{2}(x,s)dx\,ds,$

with $q^{\prime}=\frac{q}{q-1}.$ Observing that

(3.5)
$\|u\|_{2{q^{\prime}}}^{2}=\left(\int_{\mathbb{R}}(u(t))^{2{q^{\prime}}}dx\right)^{\frac{1}{q^{\prime}}}=\left(\int_{\mathbb{R}}u^{2}(t)(u(t))^{\frac{2}{q-1}}dx\right)^{\frac{1}{q^{\prime}}}\leq\|u\|_{2}^{\frac{2}{q^{\prime}}}\|u\|_{\infty}^{\frac{2}{{q^{\prime}}(q-1)}}=\|u\|_{2}^{\frac{2}{q^{\prime}}}\|u\|_{\infty}^{\frac{2}{q}}$

and using (3.5) in (3.4), yields that

(3.6)
$\displaystyle\begin{split}\frac{d{\mathcal{E}}}{dt}(t)\leq&-\|\partial^{j}_{x}u(t)\|_{2}^{2}-(\gamma_{0}-\gamma)\|u(t)\|_{2}^{2}+\|\beta\|_{q}\|u\|_{2}^{\frac{2}{q^{\prime}}}\|u(t)\|_{\infty}^{\frac{2}{q}}\\\
&-\frac{1}{2}\int_{t-\tau}^{t}\int_{\mathbb{R}}e^{-(t-s)}|\lambda(x)|u^{2}(x,s)dx\,ds.\end{split}$

Finally, note that the following inequality holds true

(3.7) $\|v\|_{\infty}^{2}\leq 2\|v\|_{2}\|\partial^{j}_{x}v\|_{2},$

for all $v\in H^{j}(\mathbb{R})$. Indeed, if $v\in C_{c}^{\infty}(\mathbb{R})$
and $y\in\mathbb{R}$, then we have the following inequality:

$\left|v(y)^{2}\right|=\left|\int_{-\infty}^{y}2vv_{x}dx\right|\leqslant
2\int_{-\infty}^{\infty}|v|\cdot\left|v_{x}\right|dx\leqslant
2\|v\|_{2}\left\|v_{x}\right\|_{2},$

since for $j\geq 1$ we have the following embedding

$H^{j}(\mathbb{R})\hookrightarrow H^{1}(\mathbb{R}),$

proving our estimate for smooth functions. The general case follows by
density. So, by Young inequality, from (3.6) and using (3.7), we have for
every fixed $\delta>0$ the following

$\begin{split}\frac{d{\mathcal{E}}}{dt}(t)\leq&-\|\partial^{j}_{x}u(t)\|_{2}^{2}-(\gamma_{0}-\gamma)\|u(t)\|_{2}^{2}+2^{1/q}\|\beta\|_{q}\|u\|_{2}^{\frac{2q-1}{q}}\|u_{x}\|_{2}^{\frac{1}{q}}\\\
&-\frac{1}{2}\int_{t-\tau}^{t}\int_{\mathbb{R}}e^{-(t-s)}|\lambda(x)|u^{2}(x,s)dxds\\\
\leq&-\|\partial^{j}_{x}u(t)\|_{2}^{2}-(\gamma_{0}-\gamma)\|u(t)\|_{2}^{2}+\Big{(}\frac{1}{\delta}\|\beta\|_{q}\|u\|_{2}^{\frac{2q-1}{q}}\Big{)}\left(\delta
2^{\frac{1}{q}}\|\partial^{j}_{x}u_{x}\|_{2}^{\frac{1}{q}}\right)\\\
&-\frac{1}{2}\int_{t-\tau}^{t}\int_{\mathbb{R}}e^{-(t-s)}|\lambda(x)|u^{2}(x,s)dxds\\\
\leq&-\|\partial^{j}_{x}u(t)\|_{2}^{2}-(\gamma_{0}-\gamma)\|u(t)\|_{2}^{2}+\frac{\Big{(}\frac{1}{\delta}\|\beta\|_{q}\|u\|_{2}^{\frac{2q-1}{q}}\Big{)}^{\frac{2q}{2q-1}}}{\frac{2q}{2q-1}}+\frac{\left(\delta
2^{\frac{1}{q}}\|\partial_{x}^{j}u_{x}(t)\|_{2}^{\frac{1}{q}}\right)^{2q}}{2q}\\\
&-\frac{1}{2}\int_{t-\tau}^{t}\int_{\mathbb{R}}e^{-(t-s)}|\lambda(x)|u^{2}(x,s)dxds.\end{split}$

Pick $\delta>0$ such that $4\delta^{2q}=2q,$ which gives

$\displaystyle\frac{d{\mathcal{E}}}{dt}(t)\leq$
$\displaystyle-\Big{(}\gamma_{0}-\gamma-\frac{2q-1}{2q}\Big{(}\frac{2}{q}\Big{)}^{\frac{1}{2q-1}}\|\beta\|_{p}^{\frac{2q}{2q-1}}\Big{)}\|u(t)\|_{2}^{2}-\frac{1}{2}\int_{t-\tau}^{t}\int_{\mathbb{R}}e^{-(t-s)}|\lambda(x)|u^{2}(x,s)dxds.$

Thus, taking into account (1.13) we have

$\frac{d{\mathcal{E}}}{dt}(t)\leq-{\nu}{\mathcal{E}}(t)$

where $\nu$ is as in (1.16). To finish, estimate (3.3) is a direct consequence
of the Gronwall’s Lemma considering $C(u_{0})={\mathcal{E}}(0).$ ∎

###  3.2. Asymptotic behavior: Nonlinear system

In this section, we are interested to prove that the higher-order dispersive
system in an unbounded domain is asymptotically stable when we introduce a
damping mechanism and a delay term. Precisely, we will use the Lyapunov
approach to prove that the energy ${\mathcal{E}}(t)$ defined by (3.2) tends to
$0$ as $t$ goes to $\infty$.

Observe that we can consider the function $\lambda_{0},\lambda\in
L^{\infty}(\mathbb{R})$ satisfying (1.10), (1.12) and (1.13). Thus, in this
case, we have a simple situation that can be easily proved using the arguments
done. So, the first main result is a consequence of the previous section.

###### Proof of Theorem 1.2.

The proof of this result is a consequence of the Lyapunov functional defined
in (3.1) and is analogous to what was done in Theorem 3.1, so we will omit it.
∎

###  3.3. Asymptotic behavior: Indefinite damping

In this section, we like to see what happens with general dissipative damping.
Being precise, consider the coefficient $\lambda_{0}$ in the following system

(3.8)
$\begin{cases}u_{t}(x,t)+(-1)^{j+1}\partial_{x}^{2j+1}u(x,t)+(-1)^{m}\partial_{x}^{2m}u(x,t)+\lambda_{0}u(x,t)\\\
\hskip 113.81102pt+\lambda
u(x,t-\tau)+\frac{1}{p+1}\partial_{x}u^{p+1}(x,t)=0&\quad\text{in}\quad\mathbb{R}\times(0,\infty),\\\
u(x,s)=u_{0}(x,s)&\quad\text{in}\quad\mathbb{R}\times[-\tau,0].\end{cases}$

changing sign. Let us assume, that there exist a number $\gamma>0$ and a
function $\beta\in L^{p}(\mathbb{R})$ for some $1\leq p<\infty$, such that
(1.18) and (1.19) are satisfied. In this spirit, we can prove the following
asymptotic result for the solutions of the linearized system associated with
(3.8).

######  Theorem 3.2.

Consider $\lambda,\lambda_{0}\in L^{\infty}(\mathbb{R})$, with $\lambda_{0}$
satisfying (1.18) and (1.19). If there exist a constant $\gamma>0$ and a
function $\beta\in L^{p}(\mathbb{R}),$ with the same $p$ as in (1.19), such
that the function $\lambda$ satisfies (1.12) and (1.20), so the system

(3.9)
$\begin{cases}u_{t}(x,t)+(-1)^{j+1}\partial_{x}^{2j+1}u(x,t)+(-1)^{j}\partial_{x}^{2j}u(x,t)+\lambda_{0}u(x,t)\\\
\hskip 113.81102pt+\lambda
u(x,t-\tau)=0&\quad\text{in}\quad\mathbb{R}\times(0,\infty),\\\
u(x,s)=u_{0}(x,s)&\quad\text{in}\quad\mathbb{R}\times[-\tau,0]\end{cases}$

is exponentially stable. Moreover, the solution of (3.9) satisfy

(3.10) $\color[rgb]{0,0,0}{\mathcal{E}}(t)\leq
C(u_{0},\tau)e^{-\tilde{\nu}t},$

with $\tilde{\nu}$ defined by (1.21).

###### Proof.

Taking the derivative in time of ${\mathcal{E}}(t),$ using the equation (3.9),
after that, integrating by parts and using the Young inequality we get

$\displaystyle\frac{d{\mathcal{E}}}{dt}(t)\leq$
$\displaystyle-\int_{\mathbb{R}}(\partial^{j}_{x}u)^{2}(t)dx-\gamma_{0}\int_{\mathbb{R}}u^{2}(t)dx+\int_{\mathbb{R}}(\beta_{0}(x)+\frac{e^{\tau}+1}{2}|\lambda(x)|)u^{2}(t)dx$
$\displaystyle-\frac{1}{2}\int_{t-\tau}^{t}\int_{\mathbb{R}}e^{-(t-s)}|\lambda(x)|u^{2}(x,s)dxds$
$\displaystyle\leq$
$\displaystyle-\int_{\mathbb{R}}(\partial^{j}_{x}u)^{2}(t)dx-(\gamma_{0}-\gamma)\int_{\mathbb{R}}u^{2}(t)dx+\int_{\mathbb{R}}(\beta_{0}(x)+\beta(x))u^{2}(t)dx$
$\displaystyle-\frac{1}{2}\int_{t-\tau}^{t}\int_{\mathbb{R}}e^{-(t-s)}|\lambda(x)|u^{2}(x,s)\
dx\ ds,$

Hölder inequality ensures that

(3.11)
$\frac{d{\mathcal{E}}}{dt}(t)\leq-\|\partial_{x}^{j}u(t)\|_{2}^{2}-(\gamma_{0}-\gamma)\|u(t)\|_{2}^{2}+\|\beta_{0}+\beta\|_{p}\|u\|_{2q}^{2}-\frac{1}{2}\int_{t-\tau}^{t}\int_{\mathbb{R}}e^{-(t-s)}|\lambda|u^{2}(x,s)dxds,$

with $q=\frac{p}{p-1}.$ Thanks to the inequality (3.11), we deduce that

$\displaystyle\frac{d{\mathcal{E}}}{dt}(t)\leq$
$\displaystyle-\|\partial^{j}_{x}u(t)\|_{2}^{2}-(\gamma_{0}-\gamma)\|u(t)\|_{2}^{2}+\|\beta+\beta_{0}\|_{p}\|u(t)\|_{\infty}^{\frac{2}{p}}\|u\|_{2}^{\frac{2}{q}}$
$\displaystyle-\frac{1}{2}\int_{t-\tau}^{t}\int_{\mathbb{R}}e^{-(t-s)}|\lambda|u^{2}(x,s)dxds.$

Remember that (3.7) is still valid, so thanks to the Young inequality, we
obtain

$\displaystyle\frac{d{\mathcal{E}}}{dt}(t)\leq$
$\displaystyle-\|\partial^{j}_{x}u(t)\|_{2}^{2}-(\gamma_{0}-\gamma)\|u(t)\|_{2}^{2}+\frac{\Big{(}\frac{1}{\delta}\|\beta_{0}+\beta\|_{p}\|u\|_{2}^{\frac{2p-1}{p}}\Big{)}^{\frac{2p}{2p-1}}}{\frac{2p}{2p-1}}+\frac{\left(\delta
2^{\frac{1}{p}}\|\partial^{j}_{x}u\|_{2}^{\frac{1}{p}}\right)^{2p}}{2p}$
$\displaystyle-\frac{1}{2}\int_{t-\tau}^{t}\int_{\mathbb{R}}e^{-(t-s)}|\lambda|u^{2}(x,s)dxds,$

for every fixed $\delta>0$. Now, taking $\delta>0$ such that
$4\delta^{2p}=2p,$ yields that

(3.12)
$\displaystyle\begin{split}\frac{d{\mathcal{E}}}{dt}(t)\leq&-\Big{(}\gamma-\gamma_{0}-\frac{2p-1}{2p}\Big{(}\frac{2}{p}\Big{)}^{\frac{1}{2p-1}}\|\beta_{0}+\beta\|_{p}^{\frac{2p}{2p-1}}\Big{)}\|u(t)\|_{2}^{2}\\\
&-\frac{1}{2}\int_{t-\tau}^{t}\int_{\mathbb{R}}e^{-(t-s)}|\lambda|u^{2}(x,s)dxds.\end{split}$

Finally, under the assumption (1.20) and taking in consideration (3.12), we
get

$\frac{d{\mathcal{E}}}{dt}(t)\leq-\tilde{\nu}{\mathcal{E}}(t)$

where

$\tilde{\nu}=2\Big{(}\gamma_{0}-\gamma-\frac{2p-1}{2p}\Big{(}\frac{2}{p}\Big{)}^{\frac{1}{2p-1}}\|\beta+\beta_{0}\|_{p}^{\frac{2p}{2p-1}}\Big{)}.$

This give us the exponential estimate (3.10) for the solution of (3.9), with
$\color[rgb]{0,0,0}C(u_{0},\tau)$ defined as in (1.17). ∎

Directly, we can extend the well-posedness and the stability results for the
nonlinear setting showing Theorem 1.3. This is a consequence of the previous
theorem and the results presented in Subsection 2.4.1. With this in hand, let
us prove the Corollary 1.4.

###### Proof of Corollary 1.4.

Note that, after a change of variable, the restriction of $u$ to $[t,t+T]$ is
a solution to the problem (1.2) concerning the initial data $u(t)$. Note that
$u\in C([\tau_{1},t];L^{2}({\mathbb{R}}))$ for all $\tau_{1}\in[-\tau,t)$.
Thus, by Theorem 2.9, we have

$\displaystyle\|u\|_{{\mathcal{B}}_{0,[t,t+T]}}$ $\displaystyle\leq
C_{t+T}\left\\{\|u(t)\|_{2}+\|\lambda\|_{\infty}\|u\|_{L^{1}(t-\tau,t;L^{2}(\mathbb{R}))}+\|\lambda\|_{\infty}^{1/2}\|u\|_{L^{2}(t-\tau,t;L^{2}(\mathbb{R}))}\right\\},$

where $C_{s}$ is given by

$C_{s}=\sqrt{\frac{3}{2}}\left(1+e^{2\|\lambda\|_{\infty}s}\right)^{1/2}e^{(\|\lambda\|_{\infty}+\|\lambda_{0}\|_{\infty}))s}.$

Thus, from Theorem 1.3 it follows that

$\displaystyle\|u\|_{{\mathcal{B}}_{0,[t,t+T]}}$ $\displaystyle\leq
C_{2T}\left\\{2C(u_{0},\tau)e^{-\nu
t}+\|\lambda\|_{\infty}\|u\|_{L^{1}(t-\tau,t;L^{2}(\mathbb{R}))}+\|\lambda\|_{\infty}^{1/2}\|u\|_{L^{2}(t-\tau,t;L^{2}(\mathbb{R}))}\right\\}$

where $C(u_{0},\tau)$ is given by (1.17). ∎

##  4\. Exponential stabilization: Damping system

In this section, we establish the exponential stability in the space
$H^{s}({\mathbb{R}}),$ for $s\in[0,2j+1]$, for the general dispersive system
(1.2) without the time delay term ($\tau=\lambda=0$), more precisely

(4.1)
$\begin{cases}u_{t}(x,t)+(-1)^{j+1}\partial_{x}^{2j+1}u(x,t)+(-1)^{m}\partial_{x}^{2m}u(x,t)+\lambda_{0}(x)u(x,t)\\\
\hskip
113.81102pt+{\color[rgb]{0,0,0}\frac{1}{p+1}\partial_{x}u^{p+1}(x,t)}=0&\quad\text{in}\quad\mathbb{R}\times(0,\infty),\\\
u(x,0)=u_{0}(x)&\quad\text{in}\quad\mathbb{R},\end{cases}$

with $m\leq j$, $j,m\in\mathbb{N},$ and $1\leq p<2j$.

######  Remarks 4.1.

It is important to point out that:

  * i.

The strategy to obtain the stabilization results in $H^{s}(\mathbb{R})$, for
any $s\in[0,2j+1]$, will be to prove the result in $L^{2}(\mathbb{R})$, and
after that, to prove the results in the domain of the operator, that is,
$H^{2j+1}(\mathbb{R})$. So, with these two results in hand, we employ the
interpolation results due to J.-L. Lions [25] to get the exponential decay for
any $s\in[0,2j+1]$.

  * ii.

Note that when $\tau=\lambda=0$, the stabilization for the system (4.1) in
$L^{2}(\mathbb{R}^{n})$ holds thanks to the Theorem 1.2 with localized damping
$\lambda_{0}$(see Section 3).

###  4.1. Stabilization in $H^{2j+1}({\mathbb{R}})$

Note that due to the Remarks 4.1, we will just consider the stabilization
problem associated with the solutions of (4.1) in the space
$H^{2j+1}({\mathbb{R}})$.

######  Proposition 4.2.

Let $T>0$. For $1\leq p<2j$, with $j\geq 1$ and $\lambda_{0}$ satisfying
(1.10) or (1.18), there exist $\gamma>0$, $T_{0}>0$ and a nonnegative
continuous function $\alpha_{3}:{\mathbb{R}}^{+}\rightarrow{\mathbb{R}}^{+}$,
such that, for every $u_{0}\in H^{2j+1}({\mathbb{R}})$, the corresponding
solution $u$ of (4.1) satisfies

(4.2)
$\|u(t)\|_{H^{2j+1}({\mathbb{R}})}\leq\alpha_{2j+1}(\|u_{0}\|_{2},T_{0})\|u_{0}\|_{H^{2j+1}({\mathbb{R}})}e^{-\gamma
t},\qquad\forall t\geq T_{0}.$

###### Proof.

Firstly, note that there exists a positive constant $c$ such that the
following estimate holds

$\frac{1}{c}\|u(t)\|_{H^{2j+1}({\mathbb{R}})}\leq\|u(t)\|_{2}+\|\partial_{x}^{2j+1}u(t)\|_{2},$

thus, from Theorem 1.3, it follows that

(4.3) $\frac{1}{c}\|u(t)\|_{H^{2j+1}({\mathbb{R}})}\leq
2C(u_{0},\tau)e^{-\tilde{\nu}t}+\|\partial_{x}^{2j+1}u(t)\|_{2},\quad\text{for
all $t>0$},$

with $\tilde{\nu}$ defined by (1.21) and $C(u_{0},\tau)>0$ as in (1.17). Since
$\tau=0$, then $C(u_{0},0)=\frac{1}{2}\|u_{0}\|_{2}^{2}$. The inequality (4.3)
shows that we just need to establish an exponential estimate for the $2j+1$
derivative in space of $u(t)$. To do that, observe that

$\|\partial_{x}^{2j+1}u(t)\|_{2}\leq\|u_{t}(t)\|+\|\partial_{x}^{2m}u(t)\|_{2}+\|u^{p}u_{x}(t)\|_{2}+\|\lambda_{0}u(t)\|_{2},\forall
t>0.$

From Gagliardo-Neirenberg inequality, it yields that

$\displaystyle\|\partial_{x}^{2j+1}u(t)\|_{2}\leq\|u_{t}(t)\|+\|\partial_{x}^{2j+1}u(t)\|_{2}^{\frac{2m}{2j+1}}\|u(t)\|_{2}^{1-\frac{2m}{2j+1}}+\|u(t)\|_{\infty}^{p}\|u_{x}(t)\|_{2}+\|\lambda_{0}\|_{\infty}\|u(t)\|_{2}.$

Young inequality together with Theorem 1.3, when $\lambda=\tau=0$, give that

$\displaystyle\frac{1}{2}\|\partial_{x}^{2j+1}u(t)\|_{2}$
$\displaystyle\leq\|u_{t}(t)\|+C\|u(t)\|_{2}+2^{\frac{p}{2}}\|u\|^{\frac{p}{2}}_{2}\|\partial_{x}u\|^{\frac{p+2}{2}}_{2}+\|\lambda_{0}\|_{\infty}\|u(t)\|_{2}$
$\displaystyle\leq\|u_{t}(t)\|+\left(C+\|\lambda_{0}\|_{\infty}\right)\|u(t)\|_{2}+2^{\frac{p}{2}}C\|\partial^{2j+1}_{x}u\|^{\frac{p+2}{2(2j+1)}}_{2}\|u\|_{2}^{\frac{p+2}{2}\left(1-\frac{1}{2j+1}\right)+\frac{p}{2}}.$

Hence, we obtain

$\displaystyle\frac{1}{4}\|\partial_{x}^{2j+1}u(t)\|_{2}$
$\displaystyle\leq\|u_{t}(t)\|+\left(C+\|\lambda_{0}\|_{\infty}\right)\|u(t)\|_{2}+C\|u\|_{2}^{\left(\frac{p+2}{2}\left(1-\frac{1}{2j+1}\right)+\frac{p}{2}\right)\left(\frac{2(2j+1)}{4j-p}\right)}$
(4.4)
$\displaystyle\leq\|u_{t}(t)\|+2^{\frac{1}{2}}C(u_{0},0)^{\frac{1}{2}}\left(C+\|\lambda_{0}\|_{\infty}\right)e^{-\frac{\tilde{\nu}}{2}t}+2^{\frac{1}{2}}C(u_{0},0)^{\frac{1}{2}}e^{-\frac{\tilde{\nu}}{2}\left(\frac{4j(p+1)+p}{4j-p}\right)t}.$

Let $v=u_{t}$. Then, by Proposition 2.10, with $\lambda=\tau=0$, $v$ solves
linearized equation (2.24) with initial data
$v_{0}(x)=-(-1)^{j+1}\partial_{x}^{2j+1}u_{0}+(-1)^{m}\partial_{x}^{2m}u_{0}-\frac{1}{p+1}\partial_{x}(u_{0}^{p+1})-\lambda_{0}(x)u_{0}$
such that

(4.5)
$\|v\|_{{\mathcal{B}}_{T}}\leq\sigma(\|u\|_{{\mathcal{B}}_{T}},T)\|v(0)\|_{2}.$

Now, after a change of variable, the restriction of $v$ to $[t,t+T]$ is a
solution of the system (2.24), considering $\lambda=\tau=0$, concerning the
initial data $v(t)$, thus

(4.6)
$\|v\|_{{\mathcal{B}}_{[t,t+T]}}\leq\sigma(\|u\|_{{\mathcal{B}}_{[t,t+T]}},T)\|v(t)\|_{2}.$

Applying Corollary 1.4, it follows that

$\displaystyle\|v\|_{{\mathcal{B}}_{[t,t+T]}}$
$\displaystyle\leq\sigma\Big{(}C_{2T}\left\\{2C(u_{0},0)e^{-\nu
t}\right\\},T\Big{)}\|v(t)\|_{2}.$

Therefore, we obtain

(4.7) $\displaystyle\|v\|_{{\mathcal{B}}_{[t,t+T]}}$
$\displaystyle\leq\gamma(u_{0},t,T)\|v(t)\|_{2},$

where $\gamma(s,t,T)=\sigma\left(2C_{2T}C(s,0)),T\right)$. On the other hand,
the solution $v$ may be written as

$v(t)=S(t)v_{0}-\int_{0}^{t}S(t-s)[u^{p}(s)v(s)]_{x}ds$

where $S(t)$ is a $C_{0}$-semigroup of contraction in $L^{2}({\mathbb{R}})$
generated by the operator associated to system (2.24). Note that
$v_{1}(t)=S(t)v_{0}$ is solution of the problem (2.24) with $u^{p}=0$. Then,
proceeding as in the proof of Theorem 3.2 with $\lambda=\tau=0$, we have

(4.8) $\|v_{1}(t)\|_{2}\leq
2^{1/2}C(v_{0},0)^{1/2}e^{-\frac{\nu}{2}t},\qquad\forall t\geq 0.$

with $\nu$ defined by (1.21). Denote

$v_{2}(t)=\int_{0}^{t}S(t-s)[u^{p}(s)v(s)]_{x}ds.$

Note that

$\displaystyle\|v_{2}(T)\|_{2}$
$\displaystyle\leq\|pu^{p-1}u_{x}v\|_{L^{1}(0,T;L^{2}({\mathbb{R}}))}+\|u^{p}v_{x}\|_{L^{1}(0,T;L^{2}({\mathbb{R}}))}.$

So, thanks to the Lemma 2.6, the following holds

(4.9) $\displaystyle\|v_{2}(T)\|_{2}$ $\displaystyle\leq
2^{\frac{p}{2}}T^{\frac{2j-p}{4j}}\|u\|_{\mathcal{B}_{0,T}}^{p}\|v\|_{\mathcal{B}_{0,T}}.$

Using (4.5), (4.8) and (4.9), we obtain

$\displaystyle\|v(T)\|_{2}$
$\displaystyle\leq\|v_{1}(T)\|_{2}+\|v_{1}(T)\|_{2}\leq
2^{1/2}C(v_{0},0)^{1/2}e^{-\frac{\nu}{2}t}+2^{\frac{p}{2}}T^{\frac{2j-p}{4j}}\|u\|_{\mathcal{B}_{0,T}}^{p}\|v\|_{\mathcal{B}_{0,T}}.$

Let us now consider a sequence $y_{n}(\cdot)=v(\cdot,nT)$ and introduce
$w_{n}(\cdot,t)=v(\cdot,t+nT)$. For $t\in[0,T]$, $w_{n}$ solves the problem

$\left\\{\begin{tabular}[]{l
l}$\partial_{t}w_{n}+(-1)^{j+1}\partial_{x}^{2j+1}w_{n}+(-1)^{m}\partial_{x}^{2m}w_{n}+[u(\cdot+nT)^{p}w_{n}]_{x}+\lambda_{0}w_{n}=0$&in
${\mathbb{R}}\times{\mathbb{R}}^{+}$\\\ $w_{n}(0)=y_{n}$&in
${\mathbb{R}}$.\end{tabular}\right.$

Observe that we can obtain for $y_{n}$ an estimate similar to the one obtained
for $v(T)$, namely

$\displaystyle\|y_{n+1}\|_{2}$
$\displaystyle=\|w_{n}(T)\|_{2}\leq\|S(T)y_{n}\|_{2}+\left\|\int_{0}^{T}S(T-s)[[u(s+nT)^{p}w_{n}(s)]_{x}ds\right\|_{2}$
$\displaystyle\leq
2^{1/2}C(v(\cdot+nT),0)^{1/2}e^{-\frac{\nu}{2}T}+2^{\frac{p}{2}}T^{\frac{2j-p}{4j}}\|u(\cdot+nT)\|_{\mathcal{B}_{0,T}}^{p}\|w_{n}\|_{\mathcal{B}_{0,T}}$
$\displaystyle\leq
2^{1/2}C(v(\cdot+nT),0)^{1/2}e^{-\frac{\nu}{2}T}+2^{\frac{p}{2}}T^{\frac{2j-p}{4j}}\|u\|_{\mathcal{B}_{0,[nT,(n+1)T]}}^{p}\|v\|_{\mathcal{B}_{0,[nT,(n+1)T]}}.$

Thus, (4.6) implies that

$\displaystyle\|y_{n+1}\|_{2}$ $\displaystyle\leq
2^{1/2}C(v(\cdot+nT),0)^{1/2}e^{-\frac{\nu}{2}T}+2^{\frac{p}{2}}T^{\frac{2j-p}{4j}}\|u\|_{\mathcal{B}_{0,[nT,(n+1)T]}}^{p}\sigma(\|u\|_{{\mathcal{B}}_{[nT,(n+1)T]}},T)\|y_{n}\|_{2}$

where

$C(v(\cdot+nT),0)=\frac{1}{2}\|y_{n}\|^{2}_{2},$

getting that

(4.10)
$\displaystyle\|y_{n+1}\|_{2}\leq\Big{(}e^{-\frac{\nu}{2}T}+2^{\frac{p}{2}}T^{\frac{2j-p}{4j}}\|u\|_{\mathcal{B}_{0,[nT,(n+1)T]}}^{p}\sigma(\|u\|_{{\mathcal{B}}_{[nT,(n+1)T]}},T)\Big{)}\|y_{n}\|_{2}.$

Moreover, we can choose $\beta>0$, small enough, such that

$e^{-\frac{\nu}{2}T}+2^{\frac{p}{2}}T^{\frac{2j-p}{4j}}\beta^{p}\beta\sigma(\beta,T)<1.$

With this choice of $\beta$, Corollary 1.4 allows us to pick $N>0$, large
enough, satisfying

$\|u\|_{0,[nT,(n+1)T]}\leq 2C_{2T}C(u_{0},0)e^{-\nu nT}\leq
2C_{2T}C(u_{0},0)e^{-\nu NT}\leq\beta,\qquad\forall n>N.$

Thus, from (4.10) we obtain the following estimate

$\|y_{n+1}\|_{2}\leq r\|y_{n}\|_{2},\qquad\forall n\geq N,\quad\text{where
$0<r<1$},$

which implies

(4.11) $\|v((n+k)T)\|_{2}\leq r^{k}\|v(nT)\|_{2},\qquad\forall n\geq N.$

Now, pick $T_{0}=NT$ and $t\geq T_{0}$. Then, there exists $k\in{\mathbb{N}}$
and $\theta\in[0,T]$, satisfying

$t=(N+k)T+\theta.$

Therefore, from (4.6) and (4.11), it is follows that

$\displaystyle\|v(t)\|_{2}$
$\displaystyle\leq\|v\|_{0,[(N+k)T,(N+k+1)T]}\leq\gamma(u_{0},t,T)\|v((N+k)T)\|_{2}$
$\displaystyle\leq\gamma(u_{0},t,T)r^{\frac{t-NT-\theta}{T}}\|v(T_{0})\|_{2}$
$\displaystyle\leq\gamma(u_{0},t,T)r^{\frac{t-NT-\theta}{T}}\sigma((\|u\|_{\mathcal{B}_{T_{0}}},T_{0}))\|v(0)\|_{2}$
$\displaystyle\leq\eta_{1}(\|u_{0}\|)e^{-\delta_{1}t}\|v_{0}\|_{2},$

where $\delta_{1}=\frac{1}{T}\ln\left(\frac{1}{r}\right)$ and
$\eta_{1}(s)=\gamma(u_{0},t,T)\sigma(\beta_{0}(s,T_{0}))r^{-(N+1)}$. So,

(4.12) $\|u_{t}(t)\|_{2}\leq
C\eta_{1}(\|u_{0}\|_{2})\|u_{0}\|_{H^{2j+1}({\mathbb{R}})}e^{-\delta_{1}t},\qquad\forall
t\geq T_{0}.$

Finally, from (4.1) and since $C(u_{0},0)^{1/2}\leq C\|u_{0}\|_{H^{2j+1}}$, we
have

(4.13)
$\begin{split}\frac{1}{4}\|\partial_{x}^{2j+1}u(t)\|_{2}\leq&C\eta_{1}(\|u_{0}\|_{2})\|u_{0}\|_{H^{2j+1}({\mathbb{R}})}e^{-\delta_{1}t}\\\
&+\left(C+\|\lambda_{0}\|_{\infty}\right)e^{-\frac{\tilde{\nu}}{2}t}\|u_{0}\|_{H^{2j+1}}+e^{-\frac{\tilde{\nu}}{2}\left(\frac{4j(p+1)+p}{4j-p}\right)t}\|u_{0}\|_{H^{2j+1}},\end{split}$

showing the of $\alpha_{2j+1}$ and the inequality (4.2) is verified. ∎

###  4.2. Stabilization in $H^{s}({\mathbb{R}})$

Lastly, let us present the proof of the third main result of this work.

###### Proof of Theorem 1.5.

From the interpolation inequality [25, page 19 - inequality (2.43)], we have

$\|u(t)\|_{H^{s}({\mathbb{R}})}=\|u(t)\|_{[L^{2}({\mathbb{R}}),H^{2j+1}({\mathbb{R}})]_{2,\frac{s}{2j+1}}}\leq
C\|u(t)\|_{2}^{1-\frac{s}{2j+1}}\|u(t)\|_{H^{2j+1}({\mathbb{R}})}^{\frac{s}{2j+1}}.$

Thus, Proposition 4.2 and Theorems 1.2 and 1.3, with $\lambda=\tau=0$, imply
the existence of $T_{0}>=$, $\nu>0$ and $\gamma>0$ such that

$\|u(t)\|_{H^{s}({\mathbb{R}})}\leq Ce^{-\left(1-\frac{s}{2j+1}\right)\nu
t}\|u_{0}\|_{2}^{\left(1-\frac{s}{2j+1}\right)}\alpha_{2j+1}^{\frac{s}{2j+1}}(\|u_{0}\|_{2},T_{0})e^{-\frac{s}{2j+1}\gamma
t},\qquad\forall t\geq T_{0}.$

Hence, (1.23) holds with

$\eta(s)=\left(1-\frac{s}{2j+1}\right)\nu+\frac{s}{2j+1}\gamma>0$

and

$\gamma(s,T_{0})=Cs^{\left(1-\frac{s}{2j+1}\right)}\alpha_{3}^{\frac{s}{2j+1}}(s,T_{0}),$

and the result is shown. ∎

##  5\. Concluding remarks

In this work, we can extend previous results that appear in the literature.
This happens because we treated a general higher-order dispersive system. So,
in terms of generality, we can consider the nonlinear general differential
operator

(5.1)
$\mathcal{V}u:=(-1)^{j+1}\partial_{x}^{2j+1}u(x,t)+(-1)^{m}\partial_{x}^{2m}u(x,t)+\frac{1}{p+1}\partial_{x}u^{p+1}(x,t),$

with $1\leq p<2j$, instead of the typical KdV equation (or co-related systems)
as is usual in the literature222One may generalize the linear operator
associated to $\mathcal{V}$ as
$\mathcal{V}u=\sum_{m=0}^{j}\alpha_{m}\partial_{x}^{2m+1}u+\sum_{m=0}^{j}\beta_{m}\partial_{x}^{2m}u,$
where $\alpha_{m},\beta_{m}\in\mathbb{R}$. However, the main analyses in the
paper are almost analogous without additional difficulties, thus the operator
(5.1) does not lose the generality in a sense of the aim of this paper. .
Thus, to be clear, we studied the asymptotic behavior of the equation (1.2)
posed on an unbounded domain $\mathbb{R}$ with the constant $\tau>0$ is the
time delay, the coefficients

$\lambda_{0}(x),\lambda(x)\in L^{\infty}(\mathbb{R})\quad\text{and}\quad
j,m\in\mathbb{N}.$

Therefore, considering $p=1$, when we have $j=m=1$, in (5.1), we recover the
result proved in [21] for the KdV-Burgers operator. Additionally, when $j=2$
and $m=1$, we have that the results of this manuscript are still valid for the
fifth order KdV-Burgers type operator. Finally, we can take, without loss of
generality, $j=m\in\mathbb{N}$ in (5.1). So, we can define Lyapunov
functionals associated with the solution of (1.2)

$E(t):=E(u(t))=\frac{1}{2}\int_{\mathbb{R}}u^{2}(x,t)dx$

and, for $\lambda\in L^{\infty}(\mathbb{R}),$

${\mathcal{E}}(t):={\mathcal{E}}(u(t))=\frac{1}{2}\int_{\mathbb{R}}u^{2}(x,t)dx+\frac{1}{2}\int_{t-\tau}^{t}\int_{\mathbb{R}}e^{-(t-s)}|\lambda(x)|u^{2}(x,s)\
dx\ ds.$

and thanks to the damping mechanism and the delay term we showed that
solutions of (1.2) satisfy

${\mathcal{E}}(t)\leq C(u_{0})e^{-\nu t}.$

Additionally to that, it is important to note that the Section 4 extend for
the operator (5.1) and the space $H^{s}$, for $s\in[0,2j+1]$ the results
proposed in [12, 14]. As well as, extend for an unbounded domain the results
showed in [11, 32] with appropriated choice of $j$ in the operator (5.1).
However, it is important to point out that considering the full system (1.2)
is still an open problem to prove the stabilization result in
$H^{2j+1}(\mathbb{R})$. Finally, let us give in order some further comments
related to the results here presented.

###  5.1. Weak versus strong damping mechanism

Observe that taking $\beta(x)=0$, in the Theorems 3.1, 1.2, 3.2 and 1.3, the
results are still valid and we do not need to use (3.7). However, for a more
general framework, we keep this term. Taking in mind that $\beta(x)\neq 0$,
additionally, we can consider $\lambda_{0}$ and $\lambda$ are constants such
that

$|\lambda|<\lambda_{0},$

and the delay $\tau$ is sufficiently small, so Theorem 1.2 give us the
exponential stability for the solution of (3.8). This is possible using the
method introduced in [27] for wave equations. In fact, in this case, we choose
a sufficiently small delay $\tau$ and a constant $\gamma$ such that

$\frac{e^{\tau}+1}{2}|\lambda|<\gamma<\lambda_{0}.$

Since now $\lambda_{0}=\gamma_{0}$ (see (1.10)), the conditions (1.12) and
(1.13) are satisfied, and the Theorem 1.2 follows. Note that the same remark
applies if

$\lambda,\lambda_{0}\in L^{\infty}(\mathbb{R})$

with $\lambda_{0}$ satisfying (1.10), and $|\lambda|<\gamma_{0}.$

With respect to the Theorems 3.2 and 1.3, for a more general framework, if
$\lambda_{0}$ satisfies (1.18) and (1.19) instead of

$\lambda_{0}(x)\geq\gamma_{0}\quad\text{for a.e.}\quad x\in\mathbb{R},$

then for the same function $\beta$ satisfying

$\frac{e^{\tau}+1}{2}|\lambda(x)|\leq\gamma+\beta(x)\quad\text{for a.e.}\quad
x\in\mathbb{R},$

we expect a smaller decay rate $\tilde{\nu}$ than $\nu$ in Theorems 1.18 and
1.19. The explanation for this is the fact that there is a “good” part
$\gamma$ of $\lambda_{0}$ that will compensate for the delay feedback and its
indefinite component $\beta_{0}$.

###  5.2. General framework

One may generalize the equation (1.2) as

$\begin{cases}\partial_{t}u(x,t)+\sum\limits_{m=0}^{j}\alpha_{m}\partial_{x}^{2m+1}u(x,t)+\sum\limits_{k=1}^{n}\beta_{k}\partial_{x}^{2k}u(x,t)+\lambda_{0}(x)u(x,t)\\\
\hskip
113.81102pt+\lambda(x)u(x,t-\tau)+\frac{1}{2}\partial_{x}u^{2}(x,t)=0&\quad\text{in}\quad\mathbb{R}\times(0,\infty),\\\
u(x,s)=u_{0}(x,s)&\quad\text{in}\quad\mathbb{R}\times[-\tau,0].\end{cases}$

where $\alpha_{m},\beta_{k}\in\mathbb{R}$ and $j,n\in\mathbb{N}$. In this
spirit, depending on the parameters $\alpha_{m}$ and $\beta_{k}$ we can
recover various delayed dispersive equations like:

  * 1.

Burgers equation ($\alpha_{m}=0$ and $n=1$),

  * 2.

KdV equation ($j=1$ and $\beta_{k}=0$),

  * 3.

Kawahara equation ($j=2$, $\alpha_{0}=1$, $\alpha_{1}=-1$ and $\beta_{k}=0$),

  * 4.

KdV–Burgers equation ($j=1$ and $n=1$),

  * 5.

Kawahara–Burgers equation ($j=2$, $\alpha_{0}=1$, $\alpha_{1}=-1$ and $n=1$),

  * 6.

Fourth-order dispersive equation ($\alpha_{m}=0$, $n=2$, $\beta_{1}=-1$ and
$\beta_{2}=1$),

and its generalizations. However, the main analyses in the paper are almost
analogous without additional difficulties, thus the equation (1.2) does not
lose the generality in a sense of the aim of this paper.

### Acknowledgments

This work was done during several visits of the authors at the Université de
Strasbourg and Universidade Federal de Pernambuco. They thank the members of
the departments for their hospitality. Moreover, the first author would like
to thank the Department of Mathematics of Virginia Tech, where the article was
finished.

## References

  * [1] B. Alvarez-Samaniego, and D. Lannes, _Large time existence for 3D water-waves and asymptotics_ , Invent. Math. 171 , no. 3, 485–541 (2008).
  * [2] K. Ammari, B. Chentouf and N. Smaoui, Qualitative study and numerical simulations for a time-delayed dispersive equation, J. Appl. Math. Comput. 66, 465–491 (2021).
  * [3] K. Ammari, B. Chentouf and N. Smaoui, Well-posedness and stability of a nonlinear time-delayed dispersive equation via the fixed point technique: A case study of no interior damping, Math Meth Appl Sci. , 45(8), 4555–4566 (2022).
  * [4] C. Amick, J. Bona and M. Schonbek, Decay of solutions of some nonlinear wave equations, J. Differential Equations, 81:1–49 (1989).
  * [5] L. Baudouin, E. Crépeau and J. Valein, Two approaches for the stabilization of nonlinear KdV equation with boundary time-delay feedback, IEEE Trans. Automat. Control, 64:1403–1414 (2019).
  * [6] J. L. Bona, T. Colin and D. Lannes, _Long wave approximations for water waves_ , Arch. Ration. Mech. Anal. 178, no. 3, 373–410 (2005).
  * [7] J. L. Bona, D. Lannes, and J.-C. Saut, _Asymptotic models for internal waves_ , J. Math. Pures Appl. (9) 89 , no. 6, 538–566 (2008).
  * [8] J. L. Bona and R.Scott, Solutions of the korteweg-de vries equation in fractional order sobolev spaces, Duke Math. J, 43, 87–99 (1976).
  * [9] J. M. Boussinesq, Thórie de l’intumescence liquide, applelée onde solitaire ou de, translation, se propageant dans un canal rectangulaire, C. R. Acad. Sci. Paris. 72, 755–759 (1871).
  * [10] R. A. Capistrano-Filho, C. Kwak and F. J. Vielma Leal, On the control issues for higher-order nonlinear dispersive equations on the circle, Nonlinear Analysis: Real World Applications, 68:103695, (2022).
  * [11] R. A. Capistrano-Filho and V. H. Gonzalez Martinez, Stabilization results for delayed fifth order KdV-type equation in a bounded domain, Mathematical Control and Related Fields. doi: 10.3934/mcrf.2023004.
  * [12] M.M. Cavalcanti, V.N. Domingos Cavalcanti, V. Komornik and J.H. Rodrigues, Global well-posedness and exponential decay rates for a KdV–Burgers equation with indefinite damping, Ann. I. H. Poincaré-AN, 31: 1079–1100 (2014).
  * [13] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, _Sharp global well-posedness for KdV and modified KdV on ${\mathbb{R}}$ and ${\mathbb{T}}$_, J. Amer. Math. Soc., 16, 705–749 (2003).
  * [14] F.A. Gallego and A.F. Pazoto, _On the well-posedness and asymptotic behavior of the generalized Korteweg–de Vries–Burgers equation_. Proc. Roy. Soc. Edinburgh Sect. A 149, no. 1, 219–260, (2019).
  * [15] J. Gorsky and A. A. Himonas, _Well-posedness of KdV with higher dispersion_ , Math. Comput. Simul. 80, 173–183 (2009).
  * [16] H. Hasimoto, _Water waves_ , Kagaku, 40, 401–408 (1970) [Japanese].
  * [17] H. Hirayama, _Local well-posedness for the periodic higher order KdV type equations_ , Nonlinear Differ. Equ. Appl., 19, 677–693 (2012).
  * [18] S. Hong and C. Kwak, Global well-posedness and nonsqueezing property for the higher-order KdV-type flow, Journal of Mathematical Analysis and Applications, 441:1, 140–166 (2016).
  * [19] T. Kato, _Low regularity well-posedness for the periodic Kawahara equation_ , Differ. Integral Equ., 25:11-12, 1011–1036 (2012).
  * [20] V. Komornik, On the stabilization of the Korteweg–de Vries equation, Bol. Soc. Parana. Mat. (3), 28, 33–48 (2010).
  * [21] V. Komornik and C. Pignotti, Well-posedness and exponential decay estimates for a Korteweg–de Vries–Burgers equation with time-delay, Nonlinear Analysis (191), 111646 (2020).
  * [22] V. Komornik and C. Pignotti, Energy decay for evolution equations with delay feedbacks. Mathematische Nachrichten 295.2 (2022): 377-394.
  * [23] V. Komornik, D.L. Russell and B.-Y. Zhang, Stabilization de l’équation de Korteweg–de Vries, C. R. Acad. Sci. Paris, Ser. I Math., 312, 841–843 (1991).
  * [24] D. J. Korteweg and G. de Vries, _On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves_ , Phil. Mag. 39, 422–443 (1895) .
  * [25] J.L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol.I, translated from French by P. Kenneth, Grundlehren Math. Wiss., vol.1 81, Springer-Verlag, NewYork, Heidelberg, 1972, xvi+357 pp.
  * [26] G.P. Menzala, C.F. Vasconcellos and E. Zuazua, Stabilization of the Korteweg–de Vries equation with localized damping, Quart. Appl. Math., 60, 11–129 (2002).
  * [27] S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45, 1561–1585 (2006).
  * [28] S. Nicaise and C. Pignotti, Exponential stability of abstract evolution equations with time delay, J. Evol. Equ., 15, 107–129 (2015).
  * [29] A.F. Pazoto, Unique continuation and decay for the Korteweg–de Vries equation with localized damping, ESAIM Control Optim. Calc. Var., 11, 473–486 (2005).
  * [30] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Vol. 44 of Applied Math. Sciences. Springer-Verlag, New York, 1983.
  * [31] L. Rosier and B.-Y. Zhang, Exact controllability and stabilizability of the Korteweg–de Vries equation,  Trans. Am. Math. Soc., 348, 3643–3672 (1996).
  * [32] J. Valein, On the asymptotic stability of the Korteweg-de Vries equation with time-delayed internal feedback, Mathematical Control & Related Fields, 12:3, 667-694 (2022).
  * [33] L. Tartar, Interpolation non linéaire et régularité, J. Funct. Anal.. 9, 469–489 (1972).
  * [34] R. Temam, Navier-Stokes equations: Theory and Numerical Analysis, North-Holland, Amsterdam, (1984).