# Heat equations with fast convection: source-type solutions and large-time
behaviour

Jørgen Endal  Department of Mathematical Sciences
Norwegian University of Science and Technology (NTNU)
N-7491, Trondheim, Norway<EMAIL_ADDRESS>https://folk.ntnu.no/jorgeen/
,  Liviu I. Ignat  Institute of Mathematics “Simion Stoilow” of the Romanian
Academy
21 Calea Grivitei Street, 010702 Bucharest, Romania
and
The Research Institute of the University of Bucharest - ICUB
University of Bucharest
90-92 Sos. Panduri, 5th District, Bucharest, Romania<EMAIL_ADDRESS>http://www.imar.ro/ lignat and  Fernando Quirós  Departamento de Matemáticas
Universidad Autónoma de Madrid (UAM)
Campus de Cantoblanco, 28049 Madrid, Spain
and
Instituto de Ciencias Matemáticas ICMAT (CSIC-UAM-UCM-UC3M)
28049 Madrid, Spain<EMAIL_ADDRESS>https://matematicas.uam.es/
fernando.quiros

###### Abstract.

We study the existence and uniqueness of source-type solutions to the Cauchy
problem for the heat equation with fast convection under certain tail control
assumptions. We allow the solutions to change sign, but we will in fact show
that they have the same sign as the initial data, which is a multiple of the
Dirac delta. As an application, we obtain the large-time behaviour of
nonnegative bounded solutions with integrable initial data to heat equations
with fast convection, covering the case of several dimensions that remained
open since the end of last century.

###### Key words and phrases:

Diffusion-convection, fast convection, source-type solutions, uniqueness,
asymptotic behaviour

###### 2020 Mathematics Subject Classification:

35B40, 35A01, 35A02, 35A08, 60J60

###### Contents

  1. 1 Introduction and main results
    1. 1.1 Source-type solutions
    2. 1.2 Application to large-time behaviour
    3. 1.3 Some open problems
    4. 1.4 Organization of the paper
  2. 2 Entropy formulation
    1. 2.1 Entropy solutions
    2. 2.2 Entropy subsolutions
  3. 3 Uniqueness of nonnegative source-type solutions
  4. 4 Source-type solutions have a sign
  5. 5 Application to large-time behaviour

_Dedicated to professor Yihong Du on the occasion of his 60th birthday, with
thanks for his friendship and admiration for his mathematics._

##  1\. Introduction and main results

###  1.1. Source-type solutions

The first goal of this paper is to study the existence and uniqueness of
solutions to the diffusion problem with convection

(1.1) $\partial_{t}u+\mathcal{L}u+\partial_{x_{N}}(|u|^{q-1}u)=0\quad\text{in
}Q:=(0,\infty)\times\mathbb{R}^{N},\qquad u(0)=M\delta_{0},\ M\neq
0\quad\text{on }\mathbb{R}^{N},$

in the _fast convection_ range $q\in(1-1/N,1)$, both for
$\mathcal{L}=-\Delta$, and for $\mathcal{L}=-\Delta_{x^{\prime}}$, where
$\Delta_{x^{\prime}}$ stands for the Laplacian in the first $N-1$ coordinates.
Here $\delta_{0}$ denotes the Dirac mass located at the origin. Solutions to
this problem, with a multiple of a Dirac delta as initial datum, are known as
_source-type_ solutions, and also as  _fundamental_ solutions. We are
interested in solutions that conserve mass along the evolution,

$\int_{\mathbb{R}^{N}}u(\cdot,t)=M\quad\text{for all }t>0.$

That is where the restriction for $q$ from below, $q>1-1/N$, comes from.

We start by considering the case of solutions that have a definite sign. We
assume, without loss of generality, that they are nonnegative, and prove
uniqueness in the framework of entropy solutions; see Section 2 for the
definition and main properties of solutions of this kind. This is the content
of our first theorem.

######  Theorem 1.1.

Let $q\in(1-1/N,1)$ and $\mathcal{L}=-\Delta$ or
$\mathcal{L}=-\Delta_{x^{\prime}}$. There exists at most one nonnegative
entropy solution $u$ of problem (1.1).

######  Remark 1.2.

When $\mathcal{L}=-\Delta_{x^{\prime}}$, the invariance of problem (1.1) under
certain scalings and the uniqueness result imply that source-type solutions
have a self-similar form,

$u(t,x^{\prime},x_{N})=t^{-\alpha}f(x^{\prime}t^{-1/2},x_{N}t^{-\beta}),\qquad\alpha=\frac{N+1}{2q},\quad\beta=\frac{N+1-q(N-1)}{2q};$

see [7] for a similar argument when $q>1$. If $q\geq(N+1)/(N-1)$, $\beta\leq
0$, which is incompatible with the initial datum being a multiple of a Dirac
mass.

When $q>1$ the above analysis was performed first for the rather special case
$N=1$ and $\mathcal{L}=-\Delta_{x^{\prime}}$, in which problem (1.1) reduces
to the scalar conservation law

(1.2) $\partial_{t}u+\partial_{x}(|u|^{q-1}u)=0,$

in [13], later for $N=1$ and $\mathcal{L}=-\Delta$ in [6], and finally for
both operators in any dimension in [7].

In the fast convection range, up to now only the special case $N=1$,
$\mathcal{L}=-\Delta_{x^{\prime}}$ has been covered; see [12, Proposition
1.1]. The proofs when $q>1$ use that the convection nonlinearity is $C^{1}$,
something that is not true in the fast convection range; hence the need of a
new approach. Our proof is inspired by our recent paper [5], where we consider
a nonlocal analogue of (1.1) in which the $\mathcal{L}$ is a stable nonlocal
diffusion operator.

We will next allow for sign changes. However, we will prove that under certain
tail-control assumption, source-type solutions have the same sign as the
constant $M$ in the initial datum. Hence, by the previous step, we have
uniqueness in this class.

######  Theorem 1.3.

Let $q\in(1-1/N,1)$, $\mathcal{L}=-\Delta$ or
$\mathcal{L}=-\Delta_{x^{\prime}}$, and $M>0$. Then, any entropy solution
$u\in L^{\infty}((0,\infty);L^{1}(\mathbb{R}^{N}))$ of problem (1.1) such that

(1.3) $\int_{|x|>r}|u(t,x^{\prime},x_{N})|\,{\rm d}x^{\prime}{\rm
d}x_{N}\rightarrow 0\text{ as }t\rightarrow 0^{+}\text{ for any }r>0,$

is nonnegative. As a corollary, there is at most one entropy solution to (1.1)
satisfying (1.3).

For $q\in(1,(N+1)/(N-1))$, $q\leq 2$, a similar analysis was performed in [3].
No such results were available up to now for the fast convection range.

######  Remark 1.4.

(a) The uniqueness results in [13, Theorem 1.1] for the scalar conservation
law (1.2), corresponding to the special case $N=1$,
$\mathcal{L}=-\Delta_{x^{\prime}}$, which are valid for any $q>1$, do not
assume a sign for the solution. Besides, they are valid for solutions having a
finite measure with a sign (which need not be a multiple of a Dirac mass) as
initial datum.

(b) As mentioned above, there are no nonnegative solutions to (1.1) when
$\mathcal{L}=-\Delta_{x^{\prime}}$ if $q\geq(N+1)/(N-1)$. However, this
nonexistence result was based on the uniqueness of source-type solutions.
Since the question of uniqueness of source-type solutions that may change sign
is still open in this range of exponents (even under the tail-control
assumption), we may not yet exclude the possibility of their existence.

(c) In dimension $N=1$ our proof of Theorem 1.3 is valid for $q\in(1,3)$ when
$\mathcal{L}=-\Delta$. Hence we are able to improve the available results,
covering the range $q\in(2,3)$; see Remark 4.1.

######  Theorem 1.5.

Let $q\in(1-1/N,1)$ and $\mathcal{L}=-\Delta$ or
$\mathcal{L}=-\Delta_{x^{\prime}}$. There exists a nonnegative entropy
solution $u$ of problem (1.1) satisfying the tail-control condition (1.3).

Existence of solutions with a tail control for $q>1$ in dimension $N=1$ was
proved in [6] under the restrictions $q\in(1,2)$ if
$\mathcal{L}=-\Delta_{x^{\prime}}$ and $q\in(1,3)$ if $\mathcal{L}=-\Delta$.
Higher dimensions $N>1$ were covered in [7], with the restrictions
$q\in(1,(N+1)/(N-1))$, $q\leq 2$.

###  1.2. Application to large-time behaviour

Our interest in solutions to (1.1) stems from the role they play in the
description of the large-time behaviour of bounded solutions to

(1.4) $\partial_{t}u-\Delta u+\partial_{x_{N}}(|u|^{q-1}u)=0\quad\text{in
}Q,\qquad u(0)=u_{0}\quad\text{on }\mathbb{R}^{N},$

with initial datum $u_{0}\in L^{1}(\mathbb{R}^{N})$. Indeed, in the slow
convection range $q>1$, the large-time behaviour of solutions to this problem
is given, if $\displaystyle M=\int_{\mathbb{R}^{N}}u_{0}\neq 0$, by

(1.5)
$t^{\max\\{\frac{N}{2},\frac{N+1}{2q}\\}(1-\frac{1}{p})}\|u(t)-U(t)\|_{L^{p}(\mathbb{R}^{N})}\to
0\quad\textup{as }t\to\infty$

for all $p\in[1,\infty)$, where $U$ is the unique very weak (if $q\geq
q_{*}:=1+\tfrac{1}{N}$) or entropy (if $q<q_{*}$) source-type solution in $Q$
with mass $M$ of

(1.6) $\displaystyle\partial_{t}U-\Delta U=0\quad$ $\displaystyle\textup{if
}q>q_{*},$ (1.7) $\displaystyle\partial_{t}U-\Delta
U+\partial_{x_{N}}(|U|^{q-1}U)=0\quad$ $\displaystyle\textup{if }q=q_{*},$
(1.8)
$\displaystyle\partial_{t}U-\Delta_{x^{\prime}}{U}+\partial_{x_{N}}(|U|^{q-1}U)=0\quad$
$\displaystyle\textup{if }q\in(1,q_{*}),$

as proved in the remarkable series of papers [8, 6, 7, 9] for nonnegative
solutions and in [3] for solutions with sign changes; see also [18].

The only result on long-term behaviour for solutions to (1.4) with $q<1$ known
to date was given in [12], where the author shows in the one-dimensional case
$N=1$, assuming also that the solution is nonnegative, that for all
$q\in(0,1)$ the function giving the limit is a source-type solution to (1.8),
as expected, since $1<q_{*}$. No results for $q<1$ were available in several
dimensions, the main obstruction being the lack of a uniqueness theory for
source-type solutions to the limit problem. Our uniqueness results will allow
us to overcome this difficulty, proving the large-time behaviour for
nonnegative solutions in any dimension if $q\in(1-1/N,1)$. We recall that the
restriction from below for $q$ is not technical, since it is required to
guarantee that the mass is conserved along the evolution,

$\int_{\mathbb{R}^{N}}u(\cdot,t)=\int_{\mathbb{R}^{N}}u_{0}\quad\text{for all
}t>0.$

######  Theorem 1.6.

Let $q\in(1-1/N,1)$ and $\mathcal{L}=-\Delta$ or
$\mathcal{L}=-\Delta_{x^{\prime}}$. Let $u\in
C([0,\infty);L^{1}(\mathbb{R}^{N}))$ be a nonnegative bounded entropy solution
to (1.4) with $u_{0}\in L^{1}(\mathbb{R}^{N})\cap L^{\infty}(\mathbb{R}^{N})$.
Let $U$ be the unique source-type solution to (1.1) with
$M=\int_{\mathbb{R}^{N}}u_{0}$. Then, for all $p\in[1,\infty)$,

$t^{\frac{N+1}{2q}(1-\frac{1}{p})}\|u(t)-U(t)\|_{L^{p}(\mathbb{R}^{N})}\to
0\quad\textup{as }t\to\infty.$

######  Remark 1.7.

The convergence result yields an alternative existence proof of a source-type
solution satisfying the tail-control assumption (1.3) for the case
$\mathcal{L}=-\Delta_{x^{\prime}}$.

The (parabolic and hyperbolic) estimates needed for compactness can be
obtained as in paper [5]. We will give a brief sketch in Section 5. We point
out that such compactness arguments are not valid for solutions with sign
changes.

###  1.3. Some open problems

Existence for $q\in(2,3)$ in dimension $N=2$.  As we have mentioned, the
existence results for source-type solutions in [7] for $N>1$ have the
restrictions $q<(N+1)/(N-1)$, $q\leq 2$. The first one has an essential
nature, at least when $\mathcal{L}=-\Delta_{x^{\prime}}$. But the second one,
stemming from the hyperbolic techniques used in the proof, might be technical.
Hence, it would be interesting whether there are source-type solutions for
$q\in(2,3)$ in dimension $N=2$, or, on the contrary, if they do not exist.

Non-existence for $q\geq(N+1)/(N-1)$.  Our guess would be that no source-type
solutions exist for $q\geq(N+1)/(N-1)$. However, existence has not yet been
excluded in this range when $\mathcal{L}=-\Delta$, or when
$\mathcal{L}=-\Delta_{x^{\prime}}$ if sign changes are allowed, even under
some tail-control assumption.

Long-time behaviour for solutions with sign changes.  The compactness
arguments that we are using here to obtain the large-time behaviour for
nonnegative solutions, borrowed from [5], are not able to deal with the
nonlinear term for solutions that change sign. New compactness arguments are
hence required. Nevertheless, if compactness were available, the required
uniqueness for the expected limit is already provided by the present paper.

In the one dimensional case, the method used in [4] to obtain the long time
behaviour of changing sign solutions, based on the compensated-compactness
techniques from [17], cannot be used here, since the nonlinearity is not
$C^{1}$. The issue of how to extend the compensated-compactness method in the
context of continuous flows remains to be investigated.

The case with integral 0.  If we were able to prove (1.5) for solutions with
sign changes, in the case $M=\int_{\mathbb{R}^{N}}u_{0}=0$, the result would
only say that

$\|u(\cdot,t)\|_{L^{p}(\mathbb{R}^{N})}=o\big{(}t^{\max\\{\frac{N}{2},\frac{N+1}{2q}\\}(1-\frac{1}{p})})\
\text{as}\ t\rightarrow\infty,$

and would give no information about the asymptotic shape of the solution. The
first step to understand the large-time behaviour in this interesting case
would be to obtain a sharp decay rate for the solution.

###  1.4. Organization of the paper

The rest of the paper is organized as follows. We gather in Section 2 some
preliminary material on entropy solutions (definition, properties, estimates)
that will be required later. Section 3 is devoted to prove the uniqueness of
nonnegative source-type solutions. In Section 4 we prove that source-type
solutions that may change sign, do not so if they satisfy the tail-control
property (1.3). Existence of source-type solutions is also discussed here.
Finally, as an application of our uniqueness results, we obtain the large time
behaviour of nonnegative solutions to problem (1.4) with bounded and
integrable initial data in Section 5.

##  2\. Entropy formulation

In this section we present the entropy framework for the problem

(2.1) $\partial_{t}u+\mathcal{L}u+\partial_{x_{N}}(f(u))=0\quad\text{in
}Q,\qquad u(0)=u_{0}\quad\text{on }\mathbb{R}^{N},$

where $f(u)=|u|^{q-1}u$, $1-\frac{1}{N}<q\neq 1$, $\mathcal{L}$ is either
$-\Delta$ or $-\Delta_{x^{\prime}}$, and $u_{0}$ is at least a finite measure,
gathering the preliminary material concerning solutions and sub- and
supersolutions that will be required later.

###  2.1. Entropy solutions

We start with the definition of entropy solution.

######  Definition 2.1 (Entropy solution).

A function $u$ is an _entropy solution_ of (2.1) if:

  1. (a)

(Regularity)
$u\in{L^{\infty}_{\textup{loc}}((0,\infty);L^{\infty}(\mathbb{R}^{N}))}$ and
$\nabla u\in L_{\textup{loc}}^{2}(Q)$.

  2. (b)

(Entropy inequality) For all $k\in\mathbb{R}$ and all $0\leq\phi\in
C_{\textup{c}}^{\infty}(Q)$,

(2.2) $\iint_{Q}\Big{(}|u-k|\partial_{t}\phi+\,{\rm
sgn}(u-k)\big{(}f(u)-f(k)\big{)}\partial_{x_{N}}\phi-|u-k|(\mathcal{L}\phi)\Big{)}\geq
0.$

  3. (c)

(Initial data in the sense of finite measures) For all $\psi\in
C_{\textup{b}}(\mathbb{R}^{N})$,

(2.3) $\operatorname*{ess\,lim}_{t\to
0^{+}}\int_{\mathbb{R}^{N}}u(t)\psi=\int_{\mathbb{R}^{N}}\psi\,{\rm d}u_{0}.$

######  Remark 2.2.

  1. (a)

If $u_{0}\in L^{1}(\mathbb{R}^{N})\cap L^{\infty}(\mathbb{R}^{N})$, condition
(2.3) is implied if the data is taken in an $L^{1}$-sense. It is then standard
to show that instead of assuming the $L^{1}$-continuity at $t=0$, we can add
the term $\int_{\mathbb{R}^{N}}|u_{0}-k|\phi(0)$ on the left-hand side of the
entropy inequality (2.2) and take $\phi\in C_{\rm c}^{\infty}(\overline{Q})$.

  2. (b)

It is also straightforward to show that classical solutions of (2.1) are
entropy solutions, and that entropy solutions are _very weak solutions_ of
that problem, i.e.:

$\iint_{Q}\big{(}u\partial_{t}\phi+f(u)\partial_{x_{N}}\phi-u(\mathcal{L}\phi)\big{)}+\int_{\mathbb{R}^{N}}\phi(0)\,{\rm
d}u_{0}=0\quad\text{for all }\phi\in C_{\textup{c}}^{\infty}(\overline{Q}).$

Let us now collect some more or less standard results for entropy solutions.
The proof can be deduced by following [14] or [1] with $\alpha=2$.

######  Lemma 2.3.

(a) Assume $p\in[1,\infty)$ and $u_{0}\in L^{\infty}(\mathbb{R}^{N})$. Then
there is _at least_ one entropy solution $u\in L^{\infty}(Q)\cap
C([0,\infty);L_{\textup{loc}}^{p}(\mathbb{R}^{N}))$ and $\nabla u\in
L_{\textup{loc}}^{2}(Q)$ of (2.1) with initial data $u_{0}$. Moreover, let
$u,\bar{u}$ be entropy solutions of (2.1) with initial data
$u_{0},\bar{u}_{0}$, respectively. Then:

  1. (i)

(Comparison principle) If $u_{0}\leq\bar{u}_{0}$ a.e. on $\mathbb{R}^{N}$,
then $u\leq\bar{u}$ a.e. in $Q$.

  2. (ii)

($L^{1}$-contraction) If $u_{0}-\bar{u}_{0}\in L^{1}(\mathbb{R}^{N})$, then,
for a.e. $t>0$,

$\int_{\mathbb{R}^{N}}|u(t)-\bar{u}(t)|\leq\int_{\mathbb{R}^{N}}|u_{0}-\bar{u}_{0}|.$

  3. (iii)

($L^{\infty}$-bound) For a.e. $(t,x)\in Q$,
$\operatorname*{ess\,inf}_{x\in\mathbb{R}^{N}}u_{0}(x)\leq
u(t,x)\leq\operatorname*{ess\,sup}_{x\in\mathbb{R}^{N}}u_{0}(x)$.

  4. (iv)

($L^{1}$-bound) For a.e. $t>0$,
$\|u(t)\|_{L^{1}(\mathbb{R}^{N})}\leq\|u_{0}\|_{L^{1}(\mathbb{R}^{N})}$.

  5. (v)

(Mass conservation) For a.e. $t>0$,

$\int_{\mathbb{R}^{N}}u(t)=\int_{\mathbb{R}^{N}}u_{0}.$

(b) Assume, in addition, $u_{0}\in L^{1}(\mathbb{R}^{N})\cap
L^{\infty}(\mathbb{R}^{N})$. Then there is _at most_ one entropy solution $u$
of (2.1) with initial data $u_{0}$, and moreover, $u\in
C([0,\infty);L^{1}(\mathbb{R}^{N}))$.

######  Remark 2.4.

By an approximation argument, we obtain that for general initial data which
are just finite measures, $u\in C((0,\infty);L^{1}(\mathbb{R}^{N}))$ and
$\int_{\mathbb{R}^{N}}u(t)=\int_{\mathbb{R}^{N}}{\rm d}u_{0}$ for all $t>0$.

As in [5], we also need some further regularity properties whose proofs follow
the lines of the mentioned paper except that $\alpha=2$.

######  Lemma 2.5.

Assume $0\leq u_{0}\in L^{1}(\mathbb{R}^{N})\cap L^{\infty}(\mathbb{R}^{N})$.
Then:

  1. (a)

(Energy estimate) For all $0<\tau<T<\infty$,

$\int_{\tau}^{T}\|\mathcal{L}^{\frac{1}{2}}u(s)\|_{L^{2}(\mathbb{R}^{N})}^{2}\,{\rm
d}s\leq\frac{1}{2}\|u(\tau)\|^{2}_{L^{2}(\mathbb{R}^{N})}.$

  2. (b)

(Preliminary estimate on the time derivative) For all smooth bounded domain
$\Omega\subset\mathbb{R}^{N}$,

$\partial_{t}u\in L^{2}((0,\infty);H^{-1}(\Omega)).$

  3. (c)

(Preliminary tail control) Let $\rho_{R}(x):=\rho(x/R)$, where $0\leq\rho\leq
1$ is a $C^{\infty}$-functions such that $\rho\equiv 0$ if $|x|\leq 1$ and
$\rho\equiv 1$ if $|x|>2$. Then, for all $R>0$ and $t>0$,

$\begin{split}&\int_{|x|>2R}u(t)\lesssim\int_{|x|>R}u_{0}+Mt\|\mathcal{L}\rho_{R}\|_{L^{\infty}(\mathbb{R}^{N})}+\int_{0}^{t}\int_{\mathbb{R}^{N}}u^{q}|\partial_{x_{N}}\rho_{R}|.\end{split}$

######  Remark 2.6.

  1. (a)

When $\mathcal{L}=-\Delta_{x^{\prime}}$,

$\|\mathcal{L}^{\frac{1}{2}}u(s)\|_{L^{2}(\mathbb{R}^{N})}^{2}=\int_{\mathbb{R}^{N}}|\nabla_{x^{\prime}}u(s)|^{2},$

which gives us no control on the derivative in the direction $x_{N}$.

  2. (b)

By an approximation argument, we obtain that for $u_{0}=M\delta_{0}$,

$\int_{|x|>2R}u(t)\lesssim
Mt\|\mathcal{L}\rho_{R}\|_{L^{\infty}(\mathbb{R}^{N})}+\int_{0}^{t}\int_{\mathbb{R}^{N}}u^{q}|\partial_{x_{N}}\rho_{R}|.$

###  2.2. Entropy subsolutions

In this subsection, we collect results on entropy sub- and supersolutions from
[14, 15]. We denote the respective positive and negative parts by
$s^{+}=\max\\{s,0\\}$ and $s^{-}=-\min\\{s,0\\}=\max\\{-s,0\\}$. So
$s=s^{+}-s^{-}$ and $|s|=s^{+}+s^{-}$.

######  Definition 2.7 (Entropy subsolution).

A function $u$ is an _entropy subsolution_ of (2.1) if:

  1. (a)

(Regularity) $u\in L^{\infty}(Q)$ and $\nabla u\in L^{2}_{\textup{loc}}(Q)$.

  2. (b)

(Entropy inequality) For all $k\in\mathbb{R}$ and all $0\leq\phi\in
C_{\textup{c}}^{\infty}(Q)$,

(2.4) $\iint_{Q}\Big{(}(u-k)^{+}\partial_{t}\phi+\,{\rm
sgn}^{+}(u-k)\big{(}f(u)-f(k)\big{)}\partial_{x_{N}}\phi-(u-k)^{+}(\mathcal{L}\phi)\Big{)}\geq
0.$

  3. (c)

(Initial data)

(2.5) $\operatorname*{ess\,lim}_{t\to
0^{+}}(u(t,x)-u_{0}(x))^{+}=0\quad\text{in
$L^{1}_{\textup{loc}}(\mathbb{R}^{N})$.}$

As before, it is possible to define the above concepts as one integral
inequality [15, Prop. 2.1]: For all $k\in\mathbb{R}$ and all $0\leq\phi\in
C_{\textup{c}}^{\infty}(\overline{Q})$

(2.6) $\iint_{Q}\Big{(}(u-k)^{+}\partial_{t}\phi+\,{\rm
sgn}^{+}(u-k)\big{(}f(u)-f(k)\big{)}\partial_{x_{N}}\phi-(u-k)^{+}(\mathcal{L}\phi)\Big{)}+\int_{\mathbb{R}^{N}}(u_{0}-k)^{+}\phi(0)\geq
0.$

In a similar way, a function $u$ is called an _entropy supersolution_ if

$\iint_{Q}\Big{(}(k-u)^{+}\partial_{t}\phi+\,{\rm
sgn}^{+}(k-u)\big{(}f(k)-f(u)\big{)}\partial_{x_{N}}\phi-(k-u)^{+}(\mathcal{L}\phi)\Big{)}+\int_{\mathbb{R}^{N}}(k-u_{0})^{+}\phi(0)\geq
0.$

Later the below properties are used on $Q$.

######  Proposition 2.8.

The following holds for equation (2.1):

  1. (a)

A function $u$ is an entropy solution iff it is an entropy sub- and
supersolution.

  2. (b)

All entropy subsolutions satisfy

$\partial_{t}u+\mathcal{L}u+\partial_{x_{N}}f(u)\leq 0\quad\text{in
$\mathcal{D}^{\prime}(Q)$}.$

  3. (c)

If $u_{1}$ and $u_{2}$ are entropy subsolutions, then
$u=\max\\{u_{1},u_{2}\\}$ is also an entropy subsolution.

  4. (d)

If there exists a unique entropy solution $u$, then all entropy sub- and
supersolutions $u_{-},u_{+}$ satisfy

$u_{-}\leq u\leq u_{+}\quad\text{a.e. in $Q$.}$

###### Proof.

The first four properties follow from [14, 15]. Also [15, Theorem 1.1] shows
that the biggest/smallest entropy solution is also the biggest/smallest
entropy sub-/supersolution. When the uniqueness of the entropy solution $u$ is
guaranteed, it satisfies the required inequality. ∎

##  3\. Uniqueness of nonnegative source-type solutions

The goal of this section is to prove the uniqueness of nonnegative solutions
to (1.1) in the fast convection range. The proof uses some ideas from the
recent paper [5], in which the authors consider nonlocal diffusion problems
with convection.

###### Proof of Theorem 1.1.

We divide the proof into several steps. Assume that (1.1) has two nonnegative
entropy solutions $u$ and $\overline{u}$ with the same initial data
$M\delta_{0}$. In view of Remark 2.4, the solutions also belong to
$L^{\infty}((0,\infty);L^{1}(\mathbb{R}^{N}))\cap
C((0,\infty);L^{1}(\mathbb{R}^{N}))$.

(i) _Integrating in the direction $x_{N}$._ It is easy to check that

(3.1) $v(t,x^{\prime}):=\int_{\mathbb{R}}u(t,x^{\prime},x_{N})\,{\rm
d}x_{N}\quad\textup{for }(t,x^{\prime})\in(0,\infty)\times\mathbb{R}^{N-1}$

belongs to $C((0,\infty);L^{1}(\mathbb{R}^{N-1}))$ and is the unique very weak
solution of $\partial_{t}v=\Delta_{x^{\prime}}v$ in
$(0,\infty)\times\mathbb{R}^{N-1}$ with initial data $M\delta_{0^{\prime}}$ in
the sense of finite measures. Thus, $v=M\Gamma_{N-1}$, where $\Gamma_{N-1}$ is
the unique fundamental solution of the equation with mass 1, which has a self-
similar form,

$\Gamma_{N-1}(t,x^{\prime})=t^{-\frac{N-1}{2}}F(t^{-\frac{1}{2}}x^{\prime}),\qquad
F(\xi)=\frac{\textrm{e}^{-|\xi|^{2}/4}}{(4\pi)^{\frac{N-1}{2}}}.$

After changing $u$ and $\overline{u}$ on a set of measure zero we then have:

$\int_{\mathbb{R}}u(t,x^{\prime},x_{N})\,{\rm
d}x_{N}=\int_{\mathbb{R}}\overline{u}(t,x^{\prime},x_{N})\,{\rm
d}x_{N}=M\Gamma_{N-1}(t,x^{\prime})\quad\textup{ {for all
}}(t,x^{\prime})\in(0,\infty)\times\mathbb{R}^{N-1}.$

(ii) _Approximation by entropy solutions with initial data in
$L^{1}(\mathbb{R}^{N})\cap L^{\infty}(\mathbb{R}^{N})$._ We start by
constructing the initial data for our approximations. Inspired by [7], for
each $r>0$, we define

$\begin{gathered}\varphi_{r}(t,x^{\prime})=\int_{-r}^{r}u(t,x^{\prime},x_{N})\,{\rm
d}x_{N}-M\Gamma_{N-1}(t,x^{\prime})=-\int_{|x_{N}|>r}u(t,x^{\prime},x_{N})\,{\rm
d}x_{N},\\\
u^{r}_{0,n}(x^{\prime},x_{N}):=\Big{(}u(1/n,x^{\prime},x_{N})-\frac{1}{2r}\varphi_{r}(1/n,x^{\prime})\Big{)}\mathds{1}_{(-r,r)}(x_{N}).\end{gathered}$

Notice that $-M\Gamma_{N-1}(t,x^{\prime})\leq\varphi_{r}(t,x^{\prime})\leq 0$.
Hence, the functions $u^{r}_{0,n}$ are nonnegative and bounded,

$\|u^{r}_{0,n}\|_{L^{\infty}(\mathbb{R}^{N})}\leq\|u(1/n)\|_{L^{\infty}(\mathbb{R}^{N})}+\frac{M}{2r}\|\Gamma_{N-1}(1/n)\|_{L^{\infty}(\mathbb{R}^{N-1})}.$

On the other hand, they have integral $M$ since

(3.2) $\int_{\mathbb{R}}u^{r}_{0,n}(x^{\prime},x_{N})\,{\rm
d}x_{N}=M\Gamma_{N-1}(1/n,x^{\prime})\quad\text{for all
}x^{\prime}\in\mathbb{R}^{N-1}.$

Let $u_{n}$ (we omit the dependence on the parameter $r$ for simplicity) be
the entropy solution of (2.1) with initial data $u_{0,n}^{r}$. Let us prove
that, for any $t>0$, the sequence $u_{n}(t)$ converges to $u(t)$ in
$L^{1}(\mathbb{R}^{N})$ as $n\to\infty$. To this aim we observe that both
$u_{n}$ and $u(\cdot+1/n)$ are entropy solutions of (2.1), whose respective
initial data, $u_{0,n}^{r}$ and $u(1/n)$, belong to $L^{1}(\mathbb{R}^{N})\cap
L^{\infty}(\mathbb{R}^{N})$. Then the $L^{1}$-contraction property shows that

$\displaystyle\big{\|}u_{n}(t)-u\big{(}t+1/n\big{)}\big{\|}_{L^{1}(\mathbb{R}^{N})}$
$\displaystyle\leq\|u^{r}_{0,n}-u(1/n)\|_{L^{1}(\mathbb{R}^{N})}$
$\displaystyle\leq\int_{\mathbb{R}^{N-1}}\bigg{(}\int_{|x_{N}|>r}u(1/n,x^{\prime},x_{N})\,{\rm
d}x_{N}+|\varphi_{r}(1/n,x^{\prime})|\bigg{)}\,{\rm d}x^{\prime}$
$\displaystyle\leq
2\int_{\mathbb{R}^{N-1}}\int_{|x_{N}|>r}u(1/n,x^{\prime},x_{N})\,{\rm
d}x_{N}{\rm d}x^{\prime}\rightarrow 0\quad\text{as $n\rightarrow\infty$.}$

Since $u\in C((0,\infty);L^{1}(\mathbb{R}^{N}))$, we have that $u(t+1/n)$
converges to $u(t)$ in $L^{1}(\mathbb{R}^{N})$ as $n\rightarrow\infty$.
Therefore, $u_{n}(t)\rightarrow u(t)$ in $L^{1}(\mathbb{R}^{N})$ as
$n\rightarrow\infty$. The contraction property implies that $u_{n}\rightarrow
u$ in $C([\tau,\infty);L^{1}(\mathbb{R}^{N}))$ as $n\rightarrow\infty$ for all
$\tau>0$.

Defining $\overline{u}_{0,n}^{r}$ and then $\overline{u}_{n}$ in a similar
way, we get by the same reasoning that
$\overline{u}_{n}(t)\rightarrow\overline{u}(t)$ in $L^{1}(\mathbb{R}^{N})$ as
$n\rightarrow\infty$.

(iii) _Approximation of the nonlinearity._ We consider the solution of the
problem (2.1) with initial data $u_{0,n}^{r}$ and nonlinearity

(3.3) $f_{\eta}(u)=(u^{2}+\eta)^{q/2}-\eta^{q/2},\quad\eta>0.$

The corresponding solution will depend on $\eta$. We denote it by
$u^{\eta}_{n}$. Note that $f_{\eta}$ is $C^{\infty}(\mathbb{R})$ and satisfies
the inequality

(3.4) $0\leq f_{\eta}(s)=(s^{2}+\eta)^{q/2}-\eta^{q/2}\leq s^{q}\quad\text{for
all $s\geq 0$.}$

Classical regularity results in Sobolev spaces for the heat equation [10] show
that $u^{\eta}_{n}\in
L^{p}_{\textup{loc}}((0,\infty);H^{2,p}(\mathbb{R}^{N}))$ and
$\partial_{t}u\in L^{p}_{\textup{loc}}((0,\infty);L^{p}(\mathbb{R}^{N}))$ for
all $p\in(1,\infty)$. It follows that $u^{\eta}_{n}$ belongs to the parabolic
Hölder spaces $C^{\delta,2\delta}(Q)$ for some $\delta>0$, [11, Section 8.5,
p. 117]. The regularity of $f_{\eta}$ guarantees that
$\partial_{x_{N}}(f_{\eta}(u^{\eta}_{n}))$ also belongs to
$C^{\delta,2\delta}(Q)$ and then regularity results for the inhomogeneous heat
equation [11, Theorem 8.7.3, p. 123] show that $u\in
C^{1+\delta,2+2\delta}(Q)$, so $u^{\eta}_{n}$ is a classical solution.

Besides, [5, Lemma B.17] gives that $u_{n}^{\eta}\rightarrow u_{n}$ in
$C([0,\infty);L^{1}(\mathbb{R}^{N}))$ as $\eta\to 0^{+}$. Hence, in view of
the previous step, $u_{n}^{\eta}\rightarrow u$ in
$C([\tau,\infty);L^{1}(\mathbb{R}^{N}))$ for all $\tau>0$. A similar result
holds for $\overline{u}_{n}^{\eta}$, which is defined analogously.

(iv) _The primitives are bounded solutions of the Hamilton-Jacobi equation._
It is easily checked that

$v_{n}^{\eta}(t,x^{\prime},x_{N}):=\int_{-\infty}^{x_{N}}u^{\eta}_{n}(t,x^{\prime},y_{N})\,{\rm
d}y_{N}\quad\text{for all
}(t,x^{\prime},x_{N})\in(0,\infty)\times\mathbb{R}^{N-1}\times\mathbb{R}$

is a nonnegative classical solution of the Hamilton-Jacobi equation

(3.5) $\partial_{t}v-\Delta v=f_{\eta}(\partial_{x_{N}}v)\quad\text{in }Q.$

Let us prove that it is bounded. Indeed, since $u^{\eta}_{n}\in
C([0,\infty);L^{1}(\mathbb{R}^{N}))$ is nonnegative,

(3.6)
$v_{n}^{\eta}(t,x^{\prime},x_{N})\leq\int_{\mathbb{R}}u^{\eta}_{n}(t,x^{\prime},y_{N})\,{\rm
d}y_{N}=:w_{\eta}^{n}(t,x^{\prime}),$

where the function $w_{\eta}^{n}$ belongs to
$C([0,\infty);L^{1}(\mathbb{R}^{N-1}))$ and solves

(3.7) $\partial_{t}w-\Delta_{x^{\prime}}w=0\quad\text{in
$(0,\infty)\times\mathbb{R}^{N-1}$,}\qquad
w(0)=M\Gamma_{N-1}(1/n)\quad\text{on $\mathbb{R}^{N-1}$.}$

The solution of this problem is
$w_{\eta}^{n}(t)=M\Gamma_{N-1}(t)\ast\Gamma_{N-1}(1/n)$. Hence,

$\|w_{\eta}^{n}(t)\|_{L^{\infty}(\mathbb{R}^{N-1})}\leq
M\|\Gamma_{N-1}(1/n)\|_{L^{\infty}(\mathbb{R}^{N-1})}\leq Mc_{n}$

for some positive constant $c_{n}$, and $v^{\eta}_{n}$ is bounded. The same
results, with the same bound, hold for $\overline{v}^{\eta}_{n}$, which is
defined analogously.

(v) _Comparison of the traces._ We claim that, for all
$(x^{\prime},x_{N})\in\mathbb{R}^{N-1}\times\mathbb{R}$,

(3.8) $\int_{-\infty}^{x_{N}-2r}u_{0,n}^{r}(x^{\prime},y_{N})\,{\rm
d}y_{N}\leq\int_{-\infty}^{x_{N}}\overline{u}_{0,n}^{r}(x^{\prime},y_{N})\,{\rm
d}y_{N}\leq\int_{-\infty}^{x_{N}+2r}u_{0,n}^{r}(x^{\prime},y_{N})\,{\rm
d}y_{N}.$

We prove only the first inequality since the second one can be obtained
similarly. By construction, $u_{0,n}^{r}(x^{\prime},\cdot)$ is supported in
$(-r,r)$. Thus, if $x_{N}<r$, the left-hand-side term vanishes, and, hence the
inequality is true since $\overline{u}_{0,n}^{r}$ is nonnegative. If
$x_{N}>r$, using the support of $\overline{u}_{0,n}^{r}(x^{\prime},\cdot)$ and
(3.2),

$\displaystyle\int_{-\infty}^{x_{N}-2r}u_{0,n}^{r}(x^{\prime},y_{N})\,{\rm
d}y_{N}$
$\displaystyle\leq\int_{-\infty}^{\infty}u_{0,n}^{r}(x^{\prime},y_{N})\,{\rm
d}y_{N}=M\Gamma_{N-1}(1/n,x^{\prime})=\int_{-\infty}^{\infty}\overline{u}_{0,n}^{r}(x^{\prime},y_{N})\,{\rm
d}y_{N}$
$\displaystyle=\int_{-\infty}^{r}\overline{u}_{0,n}^{r}(x^{\prime},y_{N})\,{\rm
d}y_{N}=\int_{-\infty}^{x_{N}}\overline{u}_{0,n}^{r}(x^{\prime},y_{N})\,{\rm
d}y_{N},$

and the inequality is also true.

On the other hand, since $u^{\eta}_{n}\in
C([0,\infty);L^{1}(\mathbb{R}^{N}))$, then $u^{\eta}_{n}(t)\rightarrow
u_{0,n}^{r}$ in $L^{1}(\mathbb{R}^{N})$ as $t\rightarrow 0^{+}$. Thus,

$v_{n}^{\eta}(t,x^{\prime},x_{N})\rightarrow\int_{-\infty}^{x_{N}}u_{0,n}^{r}(x^{\prime},y_{N})\,{\rm
d}y_{N}\quad\text{in
}L^{1}_{x^{\prime}}(\mathbb{R}^{N-1};L_{x_{N}}^{\infty}(\mathbb{R}))\text{ as
}t\rightarrow 0^{+},$

and, hence, a.e. in $\mathbb{R}^{N-1}\times\mathbb{R}$ as well. Similar
arguments hold for $\overline{v}^{\eta}_{n}$. Therefore, (3.8) translates into

(3.9)
$v^{\eta}_{n}(0+,x^{\prime},x_{N}-2r)\leq\overline{v}^{\eta}_{n}(0+,x^{\prime},x_{N})\leq
v^{\eta}_{n}(0+,x^{\prime},x_{N}+2r)\quad\text{for a.e}\
(x^{\prime},x_{N})\in\mathbb{R}^{N-1}\times\mathbb{R}.$

(vi) _Comparison of the primitives._ Let us now show that the inequalities
(3.9) for the traces imply

(3.10)
$v^{\eta}_{n}(t,x^{\prime},x_{N}-2r)\leq\overline{v}^{\eta}_{n}(t,x^{\prime},x_{N})\leq
v^{\eta}_{n}(t,x^{\prime},x_{N}+2r)\quad\text{for all
}(t,x^{\prime},x_{N})\in(0,\infty)\times\mathbb{R}^{N-1}\times\mathbb{R}.$

We only prove the first inequality, since the proof of the second is
analogous. To this aim, we define

$g(t,x^{\prime},x_{N}):=v^{\eta}_{n}(t,x^{\prime},x_{N}-2r)-\overline{v}^{\eta}_{n}(t,x^{\prime},x_{N})\quad\text{for
all }(t,x^{\prime},x_{N})\in(0,\infty)\times\mathbb{R}^{N-1}\times\mathbb{R}.$

By step (iv), $g$ is uniformly bounded by $2Mc_{n}$. Moreover, $g\in
C^{1+\delta,2+2\delta}(Q)$ and satisfies

(3.11) $\displaystyle\partial_{t}g-\Delta
g=a(t,x)\partial_{x_{N}}g\quad\text{in }Q,\qquad\lim_{t\to 0^{+}}g(t)\leq
0\quad\text{a.e. on }\mathbb{R}^{N},$

where $|a(t,x)|\leq c_{\eta,q}$.

Following [2, Theorem 3], we consider
$\psi:\mathbb{R}^{N}\rightarrow\mathbb{R}_{+}$ smooth such that $\psi(x)=0$
for $|x|\leq 1$, $\psi(x)>\|g\|_{L^{\infty}(Q)}$ for $|x|\geq 2$, with
$\psi,\nabla\psi$ and $D^{2}\psi$ bounded in $\mathbb{R}^{N}$. With this
choice we introduce, for all $\beta>0$, $\psi_{\beta}(x)=\psi(\beta x)$. It
follows that $\psi_{\beta}(x)>\|g\|_{L^{\infty}(Q)}$ for $|x|\geq 2/\beta$,
and $\nabla\psi_{\beta}$, $D^{2}\psi_{\beta}$, and $\mathcal{L}\psi_{\beta}$
all go to zero uniformly on $\mathbb{R}^{N}$ when $\beta\rightarrow 0^{+}$.
Let $\tilde{\delta}>0$ and $h(t,x):=g(t,x)-\tilde{\delta}t-\psi_{\beta}(x)$.
Then $h^{+}(t)\in W^{1,1}(\mathbb{R}^{N})$ is compactly supported in the ball
$|x|<2/\beta$ and satisfies $\nabla h^{+}(t)=\nabla h(t)\mathds{1}_{(h(t)>0)}$
for all $t>0$. Hence, by (3.11),

$\displaystyle\frac{1}{2}\frac{\rm d}{{\rm d}t}$
$\displaystyle\int_{\mathbb{R}^{N}}(h^{+}(t))^{2}=\int_{\mathbb{R}^{N}}h^{+}(t)\partial_{t}h(t)=\int_{\mathbb{R}^{N}}h^{+}(t)(\partial_{t}g(t)-\tilde{\delta})$
$\displaystyle=\int_{\mathbb{R}^{N}}h^{+}(t)(\Delta
g(t)+a\partial_{x_{N}}g(t)-\tilde{\delta})$
$\displaystyle=\int_{\mathbb{R}^{N}}h^{+}(t)\big{(}\Delta
h(t)+a\partial_{x_{N}}h(t)\big{)}+\int_{\mathbb{R}^{N}}h^{+}(t)\big{(}\Delta\psi_{\beta}+a\partial_{x_{N}}\psi_{\beta}-\tilde{\delta}\big{)}.$

Observe that for $\beta$ small enough, $\beta<\beta(\tilde{\delta})$, we have

$\big{|}\Delta\psi_{\beta}+a\partial_{x_{N}}\psi_{\beta}\big{|}<\tilde{\delta}\quad\text{in
}\mathbb{R}^{N}.$

Hence, for all $t>0$, Young’s inequality gives

$\displaystyle\frac{1}{2}\frac{\rm d}{{\rm
d}t}\int_{\mathbb{R}^{N}}(h^{+}(t))^{2}$
$\displaystyle\leq-\int_{\mathbb{R}^{N}}|\nabla
h^{+}|^{2}+\int_{\mathbb{R}^{N}}ah^{+}(t)\partial_{x_{N}}h^{+}(t)$
$\displaystyle\leq-\int_{\mathbb{R}^{N}}|\nabla
h^{+}|^{2}+\frac{c^{2}_{n,q}}{4}\int_{\mathbb{R}^{N}}(h^{+}(t))^{2}+\int_{\mathbb{R}^{N}}|\partial_{x_{N}}h^{+}(t)|^{2}\leq\frac{c^{2}_{n,q}}{4}\int_{\mathbb{R}^{N}}(h^{+}(t))^{2}.$

Then for all $0<s<t$ we have

$\int_{\mathbb{R}^{N}}(h^{+}(t))^{2}\leq\int_{\mathbb{R}^{N}}(h^{+}(s))^{2}+\frac{c^{2}_{n,q}}{4}\int_{s}^{t}\int_{\mathbb{R}^{N}}(h^{+})^{2}.$

Observing that

$\lim_{s\rightarrow
0^{+}}\int_{\mathbb{R}^{N}}(h^{+}(s))^{2}=\lim_{s\rightarrow
0^{+}}\int_{|x|<2/\beta}\big{(}(g(s)-\tilde{\delta}s-\psi_{\beta}\big{)}^{+})^{2}=0,$

we conclude that $h^{+}\equiv 0$, by Grönwall’s inequality, so that

$g(t,x)\leq\tilde{\delta}t+\psi_{\beta}(x)\quad\text{for all }t>0,\text{ all
}x\in\mathbb{R}^{N},\text{ and all }\beta<\beta(\tilde{\delta}).$

This implies that $g(t,x)\leq\tilde{\delta}t$ on $\\{|x|<1/\beta\\}$ and,
hence, on $\mathbb{R}^{N}$ after letting $\beta\rightarrow 0^{+}$. Finally,
the result follows by taking the limit $\tilde{\delta}\rightarrow 0^{+}$.

(vii) _Conclusion._ Inequalities (3.10) can be written as

$\int_{-\infty}^{x_{N}-2r}u_{n}^{\eta}(t,x^{\prime},y_{N})\,{\rm
d}y_{N}\leq\int_{-\infty}^{x_{N}}\overline{u}_{n}^{\eta}(t,x^{\prime},y_{N})\,{\rm
d}y_{N}\leq\int_{-\infty}^{x_{N}+2r}u_{n}^{\eta}(t,x^{\prime},y_{N})\,{\rm
d}y_{N}.$

Since $u_{n}^{\eta}\rightarrow u_{n}$ in
$C([\tau,\infty);L^{1}(\mathbb{R}^{N}))$ for all $\tau>0$, we deduce that
$u_{n}^{\eta}(t,x^{\prime},\cdot)\rightarrow u_{n}(t,x^{\prime},\cdot)$ in
$L^{1}(\mathbb{R})$ for a.e. $x^{\prime}\in\mathbb{R}^{N-1}$ and all
$t\geq\tau$. Then, letting $\eta\rightarrow 0$ we get, up to a subsequence,

$\int_{-\infty}^{x_{N}-2r}u_{n}(t,x^{\prime},y_{N})\,{\rm
d}y_{N}\leq\int_{-\infty}^{x_{N}}\overline{u}_{n}(t,x^{\prime},y_{N})\,{\rm
d}y_{N}\leq\int_{-\infty}^{x_{N}+2r}u_{n}(t,x^{\prime},y_{N})\,{\rm
d}y_{N}\quad\text{for a.e. }x^{\prime}.$

On the other hand, since $u_{n}(t)\rightarrow u(t)$ in
$L^{1}(\mathbb{R}^{N})$, up to a subsequence,
$u_{n}(t,x^{\prime},\cdot)\rightarrow u(t,x^{\prime},\cdot)$ in
$L^{1}(\mathbb{R})$ for a.e. $x^{\prime}\in\mathbb{R}^{N-1}$. We conclude
that, for all $t\geq\tau$,

$\int_{-\infty}^{x_{N}-2r}u(t,x^{\prime},y_{N})\,{\rm
d}y_{N}\leq\int_{-\infty}^{x_{N}}\overline{u}(t,x^{\prime},y_{N})\,{\rm
d}y_{N}\leq\int_{-\infty}^{x_{N}+2r}u(t,x^{\prime},y_{N})\,{\rm
d}y_{N}\quad\text{for a.e. }x^{\prime}.$

Letting $r\to 0^{+}$, we deduce that
$u(t,x^{\prime},x_{N})=\overline{u}(t,x^{\prime},x_{N})$ for a.e.
$(x^{\prime},x_{N})\in\mathbb{R}^{N-1}\times\mathbb{R}$ and all $t\geq\tau$,
whence the uniqueness result.

_The reduced equation._ The proof is similar to the case
$\mathcal{L}=-\Delta$. Let us comment on the points where some care has to be
taken. In step (i) we still have that $v$ defined by (3.1) is the unique
fundamental solution with mass $M$ of $\partial_{t}v-\Delta_{x^{\prime}}v=0$.
In step (iii) we need not only to regularize the nonlinearity but also to add
a diffusion term in the $x_{N}$ variable

(3.12)
$\partial_{t}u-\Delta_{x^{\prime}}u-\varepsilon\partial_{x_{N}x_{N}}^{2}u+\partial_{x_{N}}(f_{\eta}(u))=0\quad\textup{in
}Q.$

Its solutions satisfy $u^{\varepsilon,\eta}\rightarrow u$ in
$C([\tau,\infty);L^{1}(\mathbb{R}^{N}))$ for all $\tau>0$ as
$\varepsilon,\eta\to 0$. The primitives in step (iv) satisfy the Hamilton-
Jacobi equation

$\partial_{t}v-\Delta_{x^{\prime}}v-\varepsilon\partial_{x_{N}x_{N}}^{2}v=f_{\eta}(\partial_{x_{N}}v)\quad\text{in
}Q,$

instead of (3.5). We can bound these primitives in terms of a function
$w_{\varepsilon}^{n}$ defined as in (3.6), which solves the same problem (3.7)
as the function $w_{\varepsilon}^{n}$ that was defined for the other case, and
that hence has the same bound. The proof of step (vi) also works nicely, once
we observe that $\int_{\mathbb{R}^{N}}h^{+}(\Delta_{x^{\prime}}h)\leq 0$ and
the term $-\varepsilon|\partial_{x_{N}}h^{+}|^{2}$ helps to absorb the extra
term appearing when applying Young’s inequality. ∎

##  4\. Source-type solutions have a sign

We now prove the uniqueness and existence of source-type solutions having a
tail-control.

###### Proof of Theorem 1.3.

We follow the main steps in [3] but adapting them to the concept of entropy
solutions.

(i) _$u^{+}$ and $u^{-}$ satisfy (2.4)._ Since $u$ is an entropy solution it
follows that $u^{+}$ and $u^{-}$ satisfy the regularity in Definition 2.7 away
from $t=0$, i.e. $u^{\pm}\in
L^{\infty}([\tau,\infty);L^{\infty}(\mathbb{R}^{N}))$ and $\nabla u^{\pm}\in
L^{2}_{\textup{loc}}([\tau,\infty)\times\mathbb{R}^{N})$ for all $\tau>0$. Let
us prove that both of them are entropy subsolutions. Since $u$ is an entropy
solution, it satisfies inequality (2.4). Clearly, the $0$-function satisfies
the same identity. Using Proposition 2.8 we obtain that $u^{+}=\max\\{u,0\\}$
is a subsolution. In the case of the negative part we first observe that,
since $f(u)$ is an odd nonlinearity, $-u$ is again an entropy solution of the
same problem with different initial data. Using again the $0$-function, we
obtain that $u^{-}=\max\\{-u,0\\}$ is an entropy subsolution.

We emphasize that in the case of one dimensional scalar conservation laws the
fact that $u^{+}$ and $u^{-}$ are entropy subsolutions has been proved in [13,
Lemma 1.1].

(ii) _Initial data of $u^{+}$ and $u^{-}$._ Since $u(t)\rightarrow
M\delta_{0}$ as $t\rightarrow 0$ it follows that for any sequence
$(t_{n})_{n\geq 1}\rightarrow 0$, there exists a subsequence $(t_{j})_{j\geq
1}\rightarrow 0$ and two nonnegative finite measures $\mu$ and $\nu$ such that
$M\delta_{0}=\mu-\nu$ and

$u^{+}(t_{j})\rightarrow\mu,\ u^{-}(t_{j})\rightarrow\nu$

in the sense of measures. The tail control implies that the two measures must
be Dirac measures supported at $x=(x^{\prime},x_{N})=0$.

(iii) _Comparison of $u^{+},u^{-}$ with entropy solutions._ Let $m_{j}\in
C_{\textup{c}}^{\infty}(\mathbb{R}^{N})$ be a sequence of nonnegative
functions converging to $M\delta_{0}$. Let us denote by $h_{j}$ and $g_{j}$
the entropy solutions of problem (2.1) with initial data
$h_{j}(0)=u^{+}(t_{j})$ and $g_{j}(0)=u^{-}(t_{j})+m_{j}$ belonging to
$L^{1}(\mathbb{R}^{N})\cap L^{\infty}(\mathbb{R}^{N})$, respectively. The
uniqueness result in Theorem 2.3(b) guarantees the uniqueness of the two
solutions. Observe that $v_{j}(t):=u^{+}(t_{j}+t)$ is an entropy subsolution
of (2.1) with the same initial data as $h_{j}$. It follows from Proposition
2.8 that $v_{j}(t)\leq h_{j}(t)$. A similar result holds for $u^{-}.$ Hence

$0\leq u^{+}(t+t_{j})\leq h_{j}(t),\ 0\leq u_{j}^{-}(t+t_{j})\leq
g_{j}(t)\quad\text{for a.e $t>0$.}$

(iv) _Estimates for $h_{j}$ and $g_{j}$._ Since $u\in
L^{\infty}((0,\infty);L^{1}(\mathbb{R}^{N}))$ it follows that

$\int_{\mathbb{R}^{N}}u^{\pm}(t_{j})\leq\int_{\mathbb{R}^{N}}|u(t_{j},x)|dx\leq\|u\|_{L^{\infty}((0,\infty);L^{1}(\mathbb{R}^{N}))}=A<\infty$

and hence the sequences $(h_{j}(0))_{j\geq 1}$ and $(g_{j}(0))_{j\geq 1}$ are
uniformly bounded in $L^{1}(\mathbb{R}^{N})$.

Let us recall some results in [5] for nonnegative solutions $w$ of (2.1) with
initial data $0\leq w_{0}\in L^{1}(\mathbb{R}^{N})\cap
L^{\infty}(\mathbb{R}^{N})$. They conserve the mass, and when
$1-\frac{1}{N}<q\leq 2$, we have hyperbolic estimates (see the proof of
Proposition 4.4 in [5] with $\alpha=2$ and $\lambda=1$):

$\bullet$ (Time decay of $L^{p}$-norm) For all $p\in[1,\infty]$ and a.e.
$t>0$,

(4.1) $\|w(t)\|_{L^{p}(\mathbb{R}^{N})}\lesssim
t^{-\frac{N+1}{2q}(1-\frac{1}{p})}\|w_{0}\|_{L^{1}(\mathbb{R}^{N})}^{\frac{1}{q}(1-\frac{1}{p})}.$

$\bullet$ (Energy estimate) For all $0<\tau<T<\infty$,

(4.2)
$\int_{\tau}^{T}\|\mathcal{L}^{\frac{1}{2}}w(t,\cdot)\|_{L^{2}(\mathbb{R}^{N})}^{2}dt\lesssim\tau^{-\frac{N+1}{2q}}\|w_{0}\|_{L^{1}(\mathbb{R}^{N})}^{1/q}.$

$\bullet$ (Estimate on the time derivative) For all bounded domains
$\Omega\subset\mathbb{R}^{N}$,

(4.3) $\|\partial_{t}w\|_{L^{2}((\tau,T);H^{-1}(\Omega))}\lesssim
C(\tau,\|w_{0}\|_{L^{1}(\mathbb{R}^{N})}).$

We now apply the above estimates to $w_{0}=h_{j}(0)$. A similar argument will
be used for $g_{j}(0)$.

Let us first consider the case when $\mathcal{L}=-\Delta$. Since
$(h_{j}(0))_{j>0}$ is uniformly bounded in $L^{1}(\mathbb{R}^{N})$, the above
estimates (4.2), (4.3) allow us to use the classical compactness argument of
Aubin-Lions-Simon [16, Theorem 5]. Thus, up to a subsequence,
$h_{j}\rightarrow h$ in $L^{2}_{\textup{loc}}(Q)$, $h_{j}\rightarrow h$ a.e.
in $Q$, and $h_{j}\rightarrow h$ in
$C_{\textup{loc}}((0,\infty);H^{-\varepsilon}(\Omega))$ for every
$\varepsilon>0$ and $\Omega\subset\mathbb{R}^{N}$ a bounded domain.

When $\mathcal{L}=-\Delta_{x^{\prime}}$, the compactness is a little bit more
difficult since the energy estimate (4.2) is only a partial gradient:

$\int_{\tau}^{T}\|\nabla_{x^{\prime}}w(t,\cdot)\|_{L^{2}(\mathbb{R}^{N})}^{2}dt\lesssim\tau^{-\frac{N+1}{2q}}\|w_{0}\|_{L^{1}(\mathbb{R}^{N})}^{1/q}.$

However, by a Hoff/Oleĭnik-type estimate, one can deduce a uniform shift
estimate in the direction $x_{N}$ (see [5, Lemma 3.8]):

(4.4) $\displaystyle\int_{|x^{\prime}|<R^{\prime},|x_{N}|<R}$
$\displaystyle|w(t,x+(0,\xi_{N}))-w(t,x)|dx$
$\displaystyle\lesssim|\xi_{N}|R^{\prime
N-1}\Big{(}R^{-1}t^{-1-\frac{N+1}{2}\frac{q-2}{q}}\|w_{0}\|_{L^{1}(\mathbb{R}^{N})}^{\frac{2-q}{q}}+t^{-\frac{1}{q}\frac{N+1}{2}}\|w_{0}\|_{L^{1}(\mathbb{R}^{N})}^{\frac{1}{q}}\Big{)}.$

The compactness result in [5, Lemma C.2] with $\alpha=2$ gives the desired
convergence of $(h_{j})_{j\geq 1}$.

The compactness obtained above allow us to pass to the limit in the entropy
inequality (2.2)

$\iint_{Q}\Big{(}|h_{j}-k|\partial_{t}\phi+\,{\rm
sgn}(h_{j}-k)\big{(}f(h_{j})-f(k)\big{)}\partial_{x_{N}}\phi-|h_{j}-k|(\mathcal{L}\phi)\Big{)}\geq
0$

to obtain that the limit point $h$ also satisfies (2.2).

Let us now identify the initial data that $h$ takes in the sense of measures.
We use the fact that $h_{j}$ conserves the mass and it is also a very weak
solution. After a classical approximation argument we deduce that for any
$\psi\in C_{\textup{c}}^{\infty}(\mathbb{R}^{N})$ we have

$\displaystyle\Big{|}\int_{\mathbb{R}^{N}}h_{j}(t,x)\psi(x)\,{\rm
d}x-\int_{\mathbb{R}^{N}}h_{j}(0)\psi(x)\,{\rm d}x\Big{|}\leq RHS$
$\displaystyle:=\int_{0}^{t}|h_{j}||\mathcal{L}\psi|+\int_{\mathbb{R}^{N}}|h_{j}|^{q}|\partial_{x_{N}}\psi|$
$\displaystyle\leq
tA\|\mathcal{L}\psi\|_{L^{\infty}(\mathbb{R}^{N})}+\int_{0}^{t}\int_{\mathbb{R}^{N}}|h_{j}|^{q}|\partial_{x_{N}}\psi|.$

Since $q\in(1-\frac{1}{N},1)$, we use the $L^{1}$-bound in Lemma 2.3 (iv) to
get

$\int_{0}^{t}\int_{\mathbb{R}^{N}}|h_{j}|^{q}|\partial_{x_{N}}\psi|\leq\int_{0}^{t}\|h_{j}\|_{L^{1}(\mathbb{R}^{N})}^{q}\|\partial_{x_{N}}\psi\|_{L^{1/(1-q)}(\mathbb{R}^{N})}\leq
C(A)\|\partial_{x_{N}}\psi\|_{L^{1/(1-q)}(\mathbb{R}^{N})}t,$

and then the $RHS$ goes to zero as $t\rightarrow 0^{+}$ uniformly in $j$.
Letting $j\rightarrow\infty$ it follows that for all $\psi\in
C_{\textup{c}}^{\infty}(\mathbb{R}^{N})$

$\int_{\mathbb{R}^{N}}h(t,x)\psi(x)\,{\rm
d}x\rightarrow\langle\mu,\psi\rangle\quad\textup{as $t\rightarrow 0^{+}$.}$

To prove the above convergence for all $\psi\in
C_{\textup{b}}(\mathbb{R}^{N})$, we use the tail control in Lemma 2.5(c) and
the above considerations to get

(4.5) $\begin{split}\int_{|x|>2R}h_{j}(t,x)\,{\rm
d}x&\lesssim\int_{|x|>R}u^{+}(t_{j})\,{\rm
d}x+\frac{At}{R^{2}}+C(A)\frac{t}{R^{1-N(q-1)}}.\end{split}$

Letting $j\rightarrow\infty$ and using that $u^{+}(t_{j})$ converges to a
multiple of Dirac mass we get

$\int_{|x|>2R}h(t,x)\,{\rm
d}x\lesssim\frac{At}{R^{2}}+C(A)\frac{t}{R^{1-N(q-1)}}$

and then again
$\int_{\mathbb{R}^{N}}h(t)\psi\rightarrow\langle\mu,\psi\rangle$ as
$t\rightarrow 0$ for all $\psi\in C_{\textup{b}}(\mathbb{R}^{N})$.

(v) _Application of uniqueness for nonnegative solutions._ It follows that $h$
is a nonnegative entropy solution of equation (2.1) with initial data a Dirac
delta denoted by $\mu$. A similar argument holds for the sequence
$(g_{j})_{j\geq 1}$ and then its limit point $g$ is again an entropy solution
of equation (2.1) but with an initial data $\nu+M\delta_{0}=\mu$. From the
uniqueness of the nonnegative solutions with Dirac delta initial data we
deduce that $h=g$.

(vi) _Conclusion._ Using that $h=g$ and letting $t_{j}\rightarrow 0^{+}$ and
then $t\rightarrow 0^{+}$ we get $\nu=0$, $\mu=M\delta_{0}$, $u^{-}=0$,
$u=u^{+}$ and the uniqueness result for nonnegative solutions applies again.
The full details are given in [3]. ∎

######  Remark 4.1.

All the estimates in Step (iv) in the proof of Theorem 1.3 are based on the
$L^{1}$–$L^{p}$ hyperbolic estimates (4.1). When $\mathcal{L}=-\Delta$ one can
also use the parabolic $L^{1}$–$L^{p}$ estimates (see [5, Lemma 3.5] with
$\alpha=2$) which are available in the range $q>1-1/N$:

$\|w(t)\|_{L^{p}(\mathbb{R}^{N})}\lesssim
t^{-\frac{N}{2}(1-\frac{1}{p})}\|w_{0}\|_{L^{1}(\mathbb{R}^{N})}\quad\textup{for
all $p\in[1,\infty]$ and $t>0$}.$

This then gives

$\int_{|x|>2R}h(t,x)\,{\rm
d}x\lesssim\frac{At}{R^{2}}+C(A)\frac{t^{1-\frac{N}{2}(q-1)}}{R}.$

Note that the right-hand side goes to zero as $t\rightarrow 0^{+}$ for all
$q\in(1,1+2/N)$. This has already been observed in dimension one for
nonnegative solutions in [6, Section 4]. We are then able to apply the
arguments in Step (iv) also when $N=1$ and $q\in(2,3)$, a case which was not
treated in [3].

We now proceed to the existence proof.

###### Proof of Theorem 1.5.

Let us consider $m_{j}=Mj^{-N}\varphi(x/j)$, $j\geq 1$, with $\varphi$ a
nonnegative smooth function with compact support and mass one. Applying the
same arguments as above to $(g_{j})_{j\geq 1}$ we obtain the required
existence. ∎

##  5\. Application to large-time behaviour

In this section, we will treat the large-time behaviour of signed solutions of
(2.1) when $q\in(1-1/N,1)$.

###### Proof of Theorem 1.6.

(i) _The case $\mathcal{L}=-\Delta$._ We introduce the one-parameter family of
functions

$u_{\lambda}(t,x^{\prime},x_{N}):=\lambda^{\gamma}u(\lambda
t,\lambda^{1/2}x^{\prime},\lambda^{\beta}x_{N}),$

where the exponents $\gamma,\beta>0$ are such that:
$\bullet$ The mass of the solutions remains constant.
$\bullet$ The decay in the $L^{\infty}$-norm remains independent of $\lambda$.
The conservation of mass imposes

$\gamma=\frac{N-1}{2}+\beta.$

Since $1-\frac{1}{N}<q<1$, we are in the subcritical case, and we rely on
hyperbolic estimates. In particular we have (4.1) with $p=\infty$. To keep
this estimate invariant for $u_{\lambda}$, we set

$\beta=\frac{N+1-q(N-1)}{2q}.$

The large time behaviour for $u$ will follow from the behaviour as
$\lambda\to\infty$ of $u_{\lambda}$ for some fixed time $t$. Note that
$u_{\lambda}$ is an entropy solution of

(5.1)
$\partial_{t}u_{\lambda}-\Delta_{x^{\prime}}u_{\lambda}-\lambda^{1-2\beta}\partial_{x_{N}x_{N}}^{2}u_{\lambda}+\partial_{x_{N}}(u_{\lambda}^{q})=0\quad\textup{in
$Q$},\qquad u_{\lambda}(0)=u_{0,\lambda}\quad\textup{in $\mathbb{R}^{N}$},$

where
$u_{0,\lambda}(x^{\prime},x_{N}):=\lambda^{\gamma}u_{0}(\lambda^{1/2}x^{\prime},\lambda^{\beta}x_{N})$
and $1-2\beta<0$. We therefore expect that the solution of (5.1) converges to
the source-type solution of
$\partial_{t}u-\Delta_{x^{\prime}}u+\partial_{x_{N}}(u^{q})=0$ as
$\lambda\to\infty$.

To do so, we need several results concerning the sequence
$(u_{\lambda})_{\lambda>1}$: time decay of $L^{p}$-norm (cf. (4.1)),
conservation of mass (cf. Lemma 2.3(a)(v)), energy estimate (cf. (4.2)), local
estimate on the $x_{N}$-derivative (cf. (4.4)), estimate on the time
derivative (cf. (4.3)), and tail control (cf. (4.5)). This is exactly
Proposition 4.4 in [5] with $\alpha=2$. All the mentioned properties can then
be transferred to $u_{\lambda}$ by scaling.

The proof of Theorem 1.6 then follows the steps of the proof of the
subcritical case for Theorem 1.3 in [5] with $\alpha=2$. Compactness in the
form of Lemma C.2 in [5] with $\alpha=2$ is mainly a consequences of the
energy estimate for the operator $-\Delta_{x^{\prime}}$ together with the
local estimate on the $x_{N}$-derivative. This and the tail control yield that
there is a subsequence such that $u_{\lambda}(t)\to U(t)$ in
$L^{1}(\mathbb{R}^{N})$ for a.e. $t>0$ and $u_{\lambda}(t,x)\to U(t,x)$ for
a.e. $(t,x)\in Q$ as $\lambda\to\infty$. The limit equation has already been
identified, but a rigorous discussion on the stability of the entropy
formulation also follows as in [5] with $\alpha=2$. Finally, the
identification of the initial data heavily depends on the deduced tail
control.

The uniqueness (cf. Theorem 1.1) of the limit equation then guarantees that
the whole sequence $(u_{\lambda})_{\lambda>0}$ converges to the limit $U$, and
we obtain Theorem 1.6.

(ii) _The case $\mathcal{L}=-\Delta_{x^{\prime}}$._ We will have to start by
replacing $\mathcal{L}=-\Delta_{x^{\prime}}$ by
$\mathcal{L}=-\Delta_{x^{\prime}}-\varepsilon\partial_{x_{N}x_{N}}^{2}$ in
(2.1). The same considerations as above imply that $u_{\lambda}^{\varepsilon}$
solves

(5.2)
$\partial_{t}u_{\lambda}^{\varepsilon}-\Delta_{x^{\prime}}u_{\lambda}^{\varepsilon}-\lambda^{1-2\beta}\varepsilon\partial_{x_{N}x_{N}}^{2}u_{\lambda}^{\varepsilon}+\partial_{x_{N}}((u_{\lambda}^{\varepsilon})^{q})=0\quad\textup{in
$Q$},\qquad u_{\lambda}^{\varepsilon}(0)=u_{0,\lambda}\quad\textup{in
$\mathbb{R}^{N}$}.$

Again, $1-2\beta<0$, and we therefore expect the same limit problem as above
when $\lambda\to\infty$.

Repeating the previous argument, $u_{\lambda}^{\varepsilon}$ also satisfies
the mentioned properties above (uniformly in $\varepsilon$ and $\lambda$).
Again, Lemma C.2 in [5] provides compactness of
$(u_{\lambda}^{\varepsilon})_{\lambda>0,\varepsilon>0}$. Up to a subsequence,
we have that $u_{\lambda_{k}}^{\varepsilon_{k}}(t)\to U(t)$ in
$L^{1}(\mathbb{R}^{N})$ for a.e. $t>0$ and
$u_{\lambda_{k}}^{\varepsilon_{k}}(t,x)\to U(t,x)$ for a.e. $(t,x)\in Q$ as
$k\to\infty$. The proof can then be finished as before. ∎

### Acknowledgements

L. I. Ignat received funding from Romanian Ministry of Research, Innovation
and Digitization, CNCS - UEFISCDI, project number PN-III-P1-1.1-TE-2021-1539,
within PNCDI III.

F. Quirós supported by grants CEX2019-000904-S, PID2020-116949GB-I00, and
RED2018-102650-T, all of them funded by MCIN/AEI/10.13039/501100011033, and by
the Madrid Government (Comunidad de Madrid – Spain) under the multiannual
Agreement with UAM in the line for the Excellence of the University Research
Staff in the context of the V PRICIT (Regional Programme of Research and
Technological Innovation).

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