# Nonlocal complement value problem for a global in time parabolic equation

Jean-Daniel Djida  ,  Guy Fabrice Foghem Gounoue  and  Yannick Kouakep
Tchaptchie  African Institute for Mathematical Sciences (AIMS), P.O. Box 608,
Limbe Crystal Gardens,South West Region, Cameroon. Email:
<EMAIL_ADDRESS>Fakultät für Mathematik Institut für
Analysis, TU Dresden Zellescher Weg 23/25, 01217, Dresden, Germany. Email:
<EMAIL_ADDRESS>Department of SFTI-School of Chemical Engineering and
Mineral Industries(EGCIM), University of N’Gaoundéré, P.O.Box 454 N’Gaoundéré
(Cameroon). Email<EMAIL_ADDRESS>
###### Abstract.

The overreaching goal of this paper is to investigate the existence and
uniqueness of weak solution of a semilinear parabolic equation with double
nonlocality in space and in time variables that naturally arises while
modeling a biological nano-sensor in the chaotic dynamics of a polymer chain.
In fact, the problem under consideration involves a symmetric
integrodifferential operator of Lévy type and a term called the interaction
potential, that depends on the time-integral of the solution over the entire
interval of solving the problem. Owing to the Galerkin approximation, the
existence and uniqueness of a weak solution of the nonlocal complement value
problem is proven for small time under fair conditions on the interaction
potential.

###### Key words and phrases:

Nonlocal operators, Lévy operators, Parabolic equations: IVP, Weak solutions.

###### 2010 Mathematics Subject Classification:

49K20, 35S15, 47G20, 45K05, 35K90

The first author is supported by the Deutscher Akademischer Austausch
Dienst/German Academic Exchange Service (DAAD)

The second author is supported by the Deutsche Forschungsgemeinschaft/German
Research Foundation (DFG) via the Research Group 3013: “Vector-and Tensor-
Valued Surface PDEs”

The third author is supported by AIMS-Cameroon research center via the travel
grant research project: Mission de recherche/EGCIM.

##  1\. Introduction

Let $\Omega$ be an open bounded set of $\mathbb{R}^{N}(N\geq 1)$. For $T>0$,
we are interested in studying the following nonlocal complement value problem

$\displaystyle\begin{cases}\partial_{t}u+\mathscr{L}u+\displaystyle\varphi\Big{(}\int_{0}^{T}u(\cdot,\tau)\,\mathrm{d}\tau\Big{)}\,u=0&\mbox{
in }\;\Omega_{T}\coloneqq\Omega\times(0,T),\\\ u=0&\mbox{ in
}\;\Sigma\coloneqq(\mathbb{R}^{N}\setminus\Omega)\times(0,T),\\\
u(\cdot,0)=u_{0}&\mbox{ in }\;\Omega,\end{cases}$ (1.1)

where $u=u(x,t)$ is an unknown scalar function, and $\varphi$ a scalar
function that will be specified in the sequel. The initial state
$u_{0}\mathrel{\mathop{\mathchar 58\relax}}\Omega\to\mathbb{R}$ is prescribed.
Here, we restrict ourselves to a purely integrodifferential operator of Lévy
type $\mathscr{L}$, which is a particular type of nonlocal operators acting on
a measurable function $u\mathrel{\mathop{\mathchar
58\relax}}{\mathbb{R}}^{N}\to{\mathbb{R}}$ as follows

$\mathscr{L}u(x)\coloneqq\mathrm{p.v.}\int_{{\mathbb{R}}^{N}}(u(x)-u(y))\nu(x-y)\;\mathrm{d}y,\qquad(x\in{\mathbb{R}}^{N}),$
(1.2)

whenever the right hand side exists and makes sense. Here and henceforward,
the function $\nu\mathrel{\mathop{\mathchar
58\relax}}{\mathbb{R}}^{N}\setminus\\{0\\}\to[0,\infty)$ is the density of a
symmetric Lévy measure. In other words, $\nu\geq 0$ and measurable such that

$\nu(-h)=\nu(h)\quad\mbox{for all}\quad
h\in{\mathbb{R}}^{N}\quad\mbox{and}\quad\int_{{\mathbb{R}}^{N}}(1\wedge|h|^{2})\nu(h)\;\mathrm{d}h<\infty.$

Notationally, we write $a\wedge b$ to denote $\min(a,b)$ for
$a,b\in{\mathbb{R}}$. For the sake of simplicity, we also assume that $\nu$
does not vanish on sets of positive measure. To wit, $\nu$ is fully supported
on $\mathbb{R}^{N}.$ A prototypical example of an operator $\mathscr{L}$ is
the fractional Laplacian $(-\Delta)^{s}$, which is obtained by taking
$\nu(h)=C_{N,s}|h|^{-N-2s}$ for $h\neq 0$ where $s\in(0,1)$ is fixed and the
constant $C_{N,s}$ is given by

$\displaystyle
C_{N,s}=\frac{2^{2s}s\Gamma\big{(}\frac{N+2s}{2}\big{)}}{\pi^{N/2}\Gamma(1-s)}.$

The constant $C_{N,s}$ is chosen so that the Fourier relation
$\widehat{(-\Delta)^{s}}u(\xi)=|\xi|^{2s}\widehat{u}(\xi)$,
$\xi\in{\mathbb{R}}^{N}$, holds for all $u\in
C^{\infty}_{c}({\mathbb{R}}^{N})$. The fractional Laplacian is one of the most
heavily studied integrodifferential operators; see for instance [BV16, CS07,
Kwa17, DNPV12, LPGea20, Gar19, CLM20] for some basics. Additional results
related to the fractional Laplacian can be found, in the references [War15,
AW20, CRS10, DNA19]. The operator $\mathscr{L}$ in (1.2) arises naturally in
probability theory as the generator of pure Lévy stochastic processes with
jump interaction measure $\nu(h)\mathrm{d}h$. We refer interested readers to
[Sat13, App09, Ber96] for more details on Lévy processes. Recent studies of
Integro-Differential Equations(IDEs) involving nonlocal operators of the form
$\mathscr{L}$ in (1.2) can be found in [Fog20, FK22]. There exists a
substantial amount of literature on nonlocal problems involving Lévy type
operators. For example, see [FKV15, Rut18, FGKV20, FK22] for the study of
elliptic problems, see [DK20, Coz17, RO16, ROS14] for the regularity of
elliptic problems, [FK13, KS14, FRRO17] for the regularity of parabolic.

Our main result (see Theorem 4.2) consists in proving the existence and
uniqueness of a weak solution to the problem (1.1) for $T$ sufficiently small
by imposing some conditions on the initial value $u_{0}$. It should be noted
that we do not require any comparability between the operators $\mathscr{L}$
and $(-\Delta)^{s}$; in the sense that the kernels $\nu$ and $|\cdot|^{-N-2s}$
need not be comparable. Following [Sta21], we do this provided that the
potential $\varphi$ satisfies the following assumption which admits functions
$\varphi$ that are not convex and not increasing at $\infty$.

######  Assumption 1.1.

The potential $\varphi\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}\to[0,\infty)$ is a continuous non-negative function such
that $\varphi(0)=0$ and $\tau\mapsto\varphi(\tau)\tau$ is a non-decreasing
differentiable function whose derivative is bounded on every compact subset of
$\mathbb{R}$.

An interesting feature of the problem under consideration is that main
equation in (1.1) contains a nonlocal operator of Lévy type in space and a
nonlocal in time term that depends on the integral over the whole interval
$(0,T)$ on which the problem is being solved, viz., problem (1.1) has a global
memory, i.e., its depends upon the memory and the future. Note in passing that
if $t\in(0,T)$ is the current time then the memory is recorded in $(0,t)$,
while the future is recorded in $(t,T)$. The mixture of nonlocal terms
(spatial and time variables) appearing in (1.1) renders the problem somehow
fully nonlocal with global memory. For this reason, the problem (1.1) is
termed nonlocal and global in time. It is noteworthy emphasizing that, similar
analysis has been carried out in [Sta21] where the Laplace operator $-\Delta$
is used in place of the nonlocal operator $\mathscr{L}$. There are several
works in the literature that study parabolic problems with memory which
include the integral of the solution from the initial to the current time,
e.g., see [CS11, VDS15, JL08, ZH20]. To the best of our knowledge, the
existing problems with memory in the literature differ from ours. Indeed, on
the one side, in our problem, we have to deal with the nonlocality in time
variable occurred by the semilinear factor depending on the integral over the
whole interval $(0,T)$ appearing in equation (1.1) whose knowledge demands to
know the so called“future” which is $T$. Moreover, the nonlocality in time in
the system (1.1) is governed by a semilinearity due to the potential
$\varphi$. On the other side, we have to deal with the nonlocality in spatial
variables, due to the nonlocal Lévy operator $\mathscr{L}$. The main novelty
of the problem (1.1) is marked by this double nonlocality in spatial and in
time variables rendering the latter the problem somewhat challenging and of
particular interest in its own. It is worth emphasizing that the nonlocal
problem in (1.1) as well as its local analogue in space cannot be reduced to
known ones by any transformation. We refer the interested reader to [Pao95,
She91]) for the studies of some local problems with initial boundary values
with memory where the “future” appears in the data. On the other hand, we
point out that our problem arises in the local setting while modeling a
biological nanosensor in the chaotic dynamics of a polymer chain s also
called, a polymer chain in an aqueous solution. In this model, the density of
the probability that a chain occupies certain region of the underlying space
allows to describe the position of the chain segment. According to [SS17], the
probability density satisfies with a high accuracy, a certain parabolic
equation of the form (1.1) in which there is a term responsible for the
interaction of the chain of polymer segments. For more on the application of
the problem (1.1), we refer interested reader for instance to [Sta21, Sta18,
SS17] and references therein. Theoretical motivations of studying the nonlocal
model (1.1) are at least twofold. Firstly, in contrast to the local model, the
advantage of the nonlocal model (1.1) is that, it allows both smooth and non-
smooth solutions $u$ in space variables. Secondly, the nonlocal model can be
viewed as an approximation of local model. For example, given $u\in
C^{2}(\mathbb{R}^{N})\cap L^{\infty}(\mathbb{R}^{N})$ one can show that
$(-\Delta)^{s}u(x)\to-\Delta u(x)$ as $s\to 1^{-}$. More generally, we have
$\mathscr{L}_{\varepsilon}u(x)\to-\Delta u(x)$ as $\varepsilon\to 0$ (see
[Fog21b, FK22]), where $\mathscr{L}_{\varepsilon}u$ is defined as in (1.2)
with $\nu$ replaced by $\nu_{\varepsilon}$ satisfying

$\displaystyle\begin{split}\nu_{\varepsilon}\geq 0\,\,\text{ is
radial},\quad\int_{\mathbb{R}^{N}}(1\land|h|^{2})\nu_{\varepsilon}(h)\mathrm{d}h=\tfrac{1}{N},\quad\text{and
$\forall\,\,\delta>0$\,\,\,}\lim_{\varepsilon\to
0}\int_{|h|>\delta}\nu_{\varepsilon}(h)\mathrm{d}h=0\,.\end{split}$ (1.3)

If we assume that $\nu\geq 0$ is radial and satisfies
$\int_{\mathbb{R}^{N}}(1\land|h|^{2})\nu(h)\mathrm{d}h=\frac{1}{N}$ then a
remarkable example of family $(\nu_{\varepsilon})_{\varepsilon}$ satisfying
(1.3), is obtained from the rescaled version $\nu$ as follows

$\displaystyle\begin{split}\nu_{\varepsilon}(h)=\begin{cases}\varepsilon^{-N-2}\nu\big{(}h/\varepsilon\big{)}&\text{if}\leavevmode\nobreak\
\leavevmode\nobreak\ |h|\leq\varepsilon\\\
\varepsilon^{-N}|h|^{-2}\nu\big{(}h/\varepsilon\big{)}&\text{if}\leavevmode\nobreak\
\leavevmode\nobreak\ \varepsilon<|h|\leq 1\\\
\varepsilon^{-N}\nu\big{(}h/\varepsilon\big{)}&\text{if}\leavevmode\nobreak\
\leavevmode\nobreak\ |h|>1.\end{cases}\end{split}$

In [Sta18], the weak solvability of the problem is proven for the case where
$u$ is a positive bounded function and $\varphi$ is the so called Flory-
Huggins potential, i.e., is a convex increasing function that tends to
infinity as its argument approaches a certain positive value. The positiveness
is a natural requirement since $u$ is a density probability. The landmark
works in [Sta21, Sta18] demonstrate that, in the case where only the Laplace
operator is involved, the problem (1.1) is well-posed for sufficiently small
time $T$. As a matter of interest the present work takes the result in [Sta21]
to the next stage by using the generator of a pure jump stochastic process of
Lévy type, which is a symmetric nonlocal operator of the form $\mathscr{L}$,
to prove further results on weak solvability for this type of problem. The
rest of the paper is structured as follows. In Section 2, we provide some
well-known results and functions spaces which are useful in this paper. In
Section 3, we prove auxiliary results which are the milestones to prove our
core result. Finally, Section 4 is devoted to the proof of the existence and
uniqueness of a weak solution to the problem (1.1) thereby constituting the
main goal of this article. We prove the existence with the aid of the
Tychonoff fixed-point theorem and prove the uniqueness for sufficiently small
$T$.

_Acknowledgment:_ The authors thank Victor Starovoitov for helpful and
productive discussions on the proof Theorem 3.2 and Lemma 4.3.

##  2\. Notations and Preliminaries

The purpose of this section is to introduce notations and some preliminary
results. Let us collect some basics on nonlocal Sobolev-like spaces in the
$L^{2}$ setting that are generalizations of Sobolev–Slobodeckij spaces and
which will be very helpful in the sequel. Let us emphasize that, these
function spaces are tailor made for the study of complement value problems
involving symmetric Lévy operators of type $\mathscr{L}$. We refer the reader
to [Fog20] more extensive discussions on this topic.

From now on, unless otherwise stated, $\Omega\subset{\mathbb{R}}^{N}$ is an
open bounded set. We also assume that $\nu\mathrel{\mathop{\mathchar
58\relax}}{\mathbb{R}}^{N}\setminus\\{0\\}\to[0,\infty]$ has full support,
satisfies the Lévy integrability condition, i.e., $\nu\in
L^{1}({\mathbb{R}}^{N},(1\wedge|h|^{2})\mathrm{d}h)$ and is symmetric, i.e.,
$\nu(h)=\nu(-h)$ for all $h\in{\mathbb{R}}^{N}$. We define the space

$V_{\nu}(\Omega|{\mathbb{R}}^{N})\coloneqq\left\\{u\mathrel{\mathop{\mathchar
58\relax}}{\mathbb{R}}^{N}\to{\mathbb{R}}\quad\mbox{meas}\mathrel{\mathop{\mathchar
58\relax}}\mathcal{E}(u,u)<\infty\right\\},$ (2.1)

where $\mathcal{E}(\cdot,\cdot)$ is the bilinear form defined by

$\mathcal{E}(u,v)\coloneqq\frac{1}{2}\iint\limits_{\mathcal{Q}(\Omega)}(u(x)-u(y))(v(x)-v(y))\nu(x-y)\;\mathrm{d}y\;\mathrm{d}x\,,$

where $\mathcal{Q}(\Omega)$ is the cross-shaped set on $\Omega$ given by

$\mathcal{Q}(\Omega)\coloneqq(\Omega\times\Omega)\cup(\Omega\times({\mathbb{R}}^{N}\setminus\Omega))\cup(({\mathbb{R}}^{N}\setminus\Omega)\times\Omega).$

We endow the space $V_{\nu}(\Omega|{\mathbb{R}}^{N})$ with the norm

$\|u\|_{V_{\nu}(\Omega|{\mathbb{R}}^{N})}\coloneqq\Big{(}\int_{\Omega}|u(x)|^{2}\;\mathrm{d}x+\mathcal{E}(u,u)\Big{)}^{\frac{1}{2}}.$

In order to study the Dirichlet problem (1.1) we also need to define the
subspace of functions in $V_{\nu}(\Omega|{\mathbb{R}}^{N})$ that vanishes on
the complement of $\Omega$, i.e.,

$\mathbb{X}_{\nu}(\Omega|{\mathbb{R}}^{N})\mathrel{\mathop{\mathchar
58\relax}}=\left\\{u\in
V_{\nu}(\Omega|{\mathbb{R}}^{N})\mathrel{\mathop{\mathchar
58\relax}}u=0\quad\mbox{a.e.}\quad\mbox{on}\quad{\mathbb{R}}^{N}\setminus\Omega\right\\},$

where $V_{\nu}(\Omega|{\mathbb{R}}^{N})$ is defined as in (2.1). The space
$\mathbb{X}_{\nu}(\Omega|{\mathbb{R}}^{N})$ is clearly a closed subspace of
$V_{\nu}(\Omega|{\mathbb{R}}^{N})$. Furthermore, if $\partial\Omega$ is
continuous then [Fog20] smooth functions of compact support
$C_{c}^{\infty}(\Omega)$ are dense in
$\mathbb{X}_{\nu}(\Omega|{\mathbb{R}}^{N})$. In addition, we have that

$\|u\|_{\mathbb{X}_{\nu}(\Omega|{\mathbb{R}}^{N})}=\Big{(}\iint\limits_{{\mathbb{R}}^{N}{\mathbb{R}}^{N}}(u(x)-u(y))^{2}\nu(x-y)\;\mathrm{d}y\;\mathrm{d}x\Big{)}^{\frac{1}{2}}$

defines an equivalent norm on $\mathbb{X}_{\nu}(\Omega|{\mathbb{R}}^{N})$.
Indeed, in virtue of the Poincaré-Friedrichs inequality on
$\mathbb{X}_{\nu}(\Omega|{\mathbb{R}}^{N})$, there exists a constant
$C=C(N,\Omega,\nu)>0$ depending only on $N,\Omega$ and $\nu$ such that

$\|u\|_{L^{2}(\Omega)}^{2}\leq
C\|u\|^{2}_{\mathbb{X}_{\nu}(\Omega|{\mathbb{R}}^{N})}\quad\mbox{for
every}\quad u\in\mathbb{X}_{\nu}(\Omega|{\mathbb{R}}^{N}).$ (2.2)

This can be verified by observing that $\mathbb{R}^{N}\setminus
B_{R}(x)\subset\mathbb{R}^{N}\setminus\Omega$ for all $x\in\Omega$, where
$R>0$ is the diameter of $\Omega$. For
$u\in\mathbb{X}_{\nu}(\Omega|{\mathbb{R}}^{N})$, we recall that
$u=0\leavevmode\nobreak\ a.e$ on $\mathbb{R}^{N}\setminus\Omega$. Hence we
have

$\displaystyle\|u\|^{2}_{\mathbb{X}_{\nu}(\Omega|{\mathbb{R}}^{N})}$
$\displaystyle\geq
2\int_{\Omega}|u(x)|^{2}\mathrm{d}x\int_{\mathbb{R}^{N}\setminus\Omega}\nu(x-y)\mathrm{d}y$
$\displaystyle\geq
2\int_{\Omega}|u(x)|^{2}\mathrm{d}x\int_{{\mathbb{R}}^{N}\setminus
B_{R}(x)}\nu(x-y)\mathrm{d}y=2\|\nu_{R}\|_{L^{1}(\mathbb{R}^{N})}\|u\|^{2}_{L^{2}(\Omega)}.$

It suffices to take C= $(2\|\nu_{R}\|_{L^{1}({\mathbb{R}}^{N})})^{-1}$ with
$\nu_{R}=\nu\mathds{1}_{\mathbb{R}^{N}\setminus B_{R}(0)}$. According to
[JW20], the Poincaré-Friedrichs inequality (2.2) remains true if $\Omega$ is
only bounded in one direction.

######  Remark 2.1.

One can observe that the notation $V_{\nu}(\Omega|{\mathbb{R}}^{N})$ is to
emphasize that the integral of the measurable map
$(x,y)\mapsto(u(x)-u(y))^{2}\nu(x-y)$ performed over
$\Omega\times{\mathbb{R}}^{N}$ is finite. Moreover, the latter is equivalent
to that performed over $\mathcal{Q}(\Omega),$ that is,
$\mathcal{E}(u,u)\asymp\iint_{\mathcal{Q}(\Omega)}(u(x)-u(y))^{2}\nu(x-y)\mathrm{d}y\mathrm{d}x.$
From the local scenario point of view, it is fair to see the space
$V_{\nu}(\Omega|{\mathbb{R}}^{N})$ as the nonlocal replacement of the
classical Sobolev space $H^{1}(\Omega)$, whereas
$\mathbb{X}_{\nu}(\Omega|{\mathbb{R}}^{N})$ can be viewed as the nonlocal
replacement of the classical Sobolev space $H^{1}_{0}(\Omega)$.

The aforementioned spaces are Hilbert spaces. Additional, recent finds about
these function spaces and their relations with classical Sobolev spaces can be
found in [Fog20, Fog21a, Fog21b]. Let $(V_{\nu}(\Omega|{\mathbb{R}}^{N}))^{*}$
and $(\mathbb{X}_{\nu}(\Omega|{\mathbb{R}}^{N}))^{*}$ be the dual spaces of
$V_{\nu}(\Omega|{\mathbb{R}}^{N})$ and
$\mathbb{X}_{\nu}(\Omega|{\mathbb{R}}^{N})$ respectively. We have the
following continuous Gelfand triple embeddings

$\mathbb{X}_{\nu}(\Omega|{\mathbb{R}}^{N})\hookrightarrow
L^{2}(\Omega)\hookrightarrow(\mathbb{X}_{\nu}(\Omega|{\mathbb{R}}^{N}))^{*}\quad\mbox{and}\quad
V_{\nu}(\Omega|{\mathbb{R}}^{N})\hookrightarrow
L^{2}(\Omega)\hookrightarrow(V_{\nu}(\Omega|{\mathbb{R}}^{N}))^{*}.$

The next result borrowed from [Fog20, JW20, DMT18], provides sufficient
conditions under which the spaces $\mathbb{X}_{\nu}(\Omega|{\mathbb{R}}^{N})$
and $V_{\nu}(\Omega|{\mathbb{R}}^{N})$ are compactly embedded in
$L^{2}(\Omega)$.

######  Theorem 2.2.

Assume that $\nu\in L^{1}(\mathbb{R}^{N},1\land|h|^{2})$ and $\nu\not\in
L^{1}(\mathbb{R}^{N})$. If $\Omega\subset\mathbb{R}^{N}$ is open and bounded
then the embedding $\mathbb{X}_{\nu}(\Omega|{\mathbb{R}}^{N})\hookrightarrow
L^{2}(\Omega)$ is compact. Furthermore, the embedding
$V_{\nu}(\Omega|{\mathbb{R}}^{N})\hookrightarrow L^{2}(\Omega)$ is also
compact provided that in addition, $\Omega$ has a Lipschitz boundary and $\nu$
satisfies

$\displaystyle\lim_{\delta\to
0}\frac{1}{\delta^{2}}\int_{B_{\delta}(0)}|h|^{2}\nu(h)\mathrm{d}h=\infty.$
(2.3)

It is worthwhile noticing that we have the natural continuous and dense
embeddings

$\displaystyle
L^{2}(0,T;\mathbb{X}_{\nu}(\Omega|{\mathbb{R}}^{N}))\hookrightarrow
L^{2}(0,T;L^{2}(\Omega))\hookrightarrow
L^{2}(0,T;(\mathbb{X}_{\nu}(\Omega|{\mathbb{R}}^{N}))^{*}).$

Next we introduce the Sobolev type space
$H_{\nu}(0,T)\mathrel{\mathop{\mathchar
58\relax}}=H^{1}(0,T;\mathbb{X}_{\nu}(\Omega|{\mathbb{R}}^{N})$ by

$H_{\nu}(0,T)=\Big{\\{}\psi\in
L^{2}(0,T;\mathbb{X}_{\nu}(\Omega|{\mathbb{R}}^{N}))\mathrel{\mathop{\mathchar
58\relax}}\quad\partial_{t}\psi\in
L^{2}(0,T;((\mathbb{X}_{\nu}(\Omega|{\mathbb{R}}^{N}))^{*})\Big{\\}}.$

The space $H_{\nu}(0,T)$ (see [LPPS15]) is a Hilbert space endowed with the
norm given by

$\|\psi\|^{2}_{H_{\nu}(0,T)}=\|\psi\|^{2}_{L^{2}(0,T;\mathbb{X}_{\nu}(\Omega|{\mathbb{R}}^{N}))}+\|\partial_{t}\psi_{L^{2}(0,T;(\mathbb{X}_{\nu}(\Omega|{\mathbb{R}}^{N}))^{*})}.$
(2.4)

As a consequence of Theorem 2.2, we get the compact embedding (2.6) below.

######  Proposition 2.3.

Assume that $\nu\not\in L^{1}(\mathbb{R}^{N})$ and
$\Omega\subset\mathbb{R}^{N}$ is open bounded. The following are true.

  1. $(i)$

Lions-Magenes Lemma [BF13, Theorem II.5.12]$\mathrel{\mathop{\mathchar
58\relax}}$ the following embedding is continuous

$H_{\nu}(0,T)\hookrightarrow C([0,T];L^{2}(\Omega)).$ (2.5)

  2. $(ii)$

Lions-Aubin Lemma [BF13, Theorem II.5.16]$\mathrel{\mathop{\mathchar
58\relax}}$ the following embedding is compact

$H_{\nu}(0,T)\hookrightarrow L^{2}(0,T;L^{2}(\Omega)).$ (2.6)

Now we state the integration by parts formula contained in [Fog20, DROV17] for
smooth functions. Precisely for every $\phi,\psi\in
C_{c}^{\infty}(\mathbb{R}^{N})$ following nonlocal Gauss-Green formula holds
true

$\displaystyle\mathcal{E}(\phi,\psi)=\int_{\Omega}\psi\mathscr{L}\phi(x)\;\mathrm{d}x+\int_{\mathbb{R}^{N}\setminus\Omega}\psi(y)\mathcal{N}\phi(y)\;\mathrm{d}y,$
(2.7)

where, $\mathcal{N}\phi$ denotes the nonlocal normal derivative $\mathcal{N}$
of $\phi$ across the boundary of $\Omega$ with respect to $\nu$ and is defined
by

$\displaystyle\mathcal{N}\phi(x)\coloneqq\int_{\Omega}\left(\phi(x)-\phi(y)\right)\nu(x-y)\;\mathrm{d}y,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
x\in{\mathbb{R}}^{N}\setminus\Omega.$ (2.8)

With the aforementioned function spaces at hand, we are now in position to
define the notion of weak solutions to the problem (1.1).

######  Definition 2.4.

Let $\varphi$ satisfies Assumption 1.1 and $u_{0}\in L^{2}(\Omega)$. A
function $u\mathrel{\mathop{\mathchar 58\relax}}\Omega_{T}\to\mathbb{R}$ is
said to be a weak solution of problem (1.1), if

  1. $(i)$

$u\in L^{2}(0,T;\mathbb{X}_{\nu}(\Omega|{\mathbb{R}}^{N}))$ and
$\partial_{t}u\in
L^{2}(0,T;\big{(}\mathbb{X}_{\nu}(\Omega|{\mathbb{R}}^{N})\big{)}^{*})$;

  2. $(ii)$

for every $\psi\in
C^{1}_{c}\big{(}[0,T);\mathbb{X}_{\nu}(\Omega|{\mathbb{R}}^{N})\big{)}$ (i.e.,
$\psi(\cdot,T)=0$), $u$ satisfies $u(\cdot,0)=u_{0}$ and

$\displaystyle\int_{\Omega}\partial_{t}u\,\psi\,\mathrm{d}x+\mathcal{E}(u,\psi)+\int_{\Omega}\varphi(v)\,u\,\psi\mathrm{d}x\,=0\quad\text{for
all $0\leq t\leq T$}.$ (2.9)

In particular, we have

$\int_{0}^{T}\int_{\Omega}u\,\partial_{t}\psi\,\mathrm{d}x\,\mathrm{d}t+\int_{0}^{T}\mathcal{E}(u,\psi)\,\mathrm{d}t+\int_{0}^{T}\int_{\Omega}\varphi(v)\,u\,\psi\mathrm{d}x\,\mathrm{d}t=\int_{\Omega}u_{0}\psi_{0}\mathrm{d}x.$

By the density of $C_{c}^{\infty}(\Omega)$ in
$\mathbb{X}_{\nu}(\Omega|{\mathbb{R}}^{N})$, it is sufficient to take $\psi\in
C_{c}^{1}([0,T);C_{c}^{\infty}(\Omega))$ as the test functions in (2.9). Our
proof of the existence of a weak solution to the problem (1.1) relies upon the
following Tychonoff fixed-point Theorem 2.5 which is a generalization of the
Brouwer and Schauder fixed-point theorems.

######  Theorem 2.5 (Tychonoff [Tyc35]).

Let $X$ be a reflexive separable Banach space. Let $G\subset X$ be closed
convex and bounded set. Any weakly sequentially continuous map
$\pi\mathrel{\mathop{\mathchar 58\relax}}G\to G$ has a fixed point.

We emphasize that when $G$ is compact and convex, Theorem 2.5 is known as the
Schauder fixed-point theorem, while, when $X$ is of finite dimension it is
known as the Brouwer fixed-point theorem.

##  3\. Nonlocal elliptic and parabolic problem

The overreaching goal of this section is to investigate weak solutions to two
specific nonlocal problems which is of interest in the proof of our main
result. The first problem is an elliptic nonlocal problem and the second one
is a parabolic nonlocal problem.

###  3.1. Nonlocal elliptic problem

Given a measurable function $f\mathrel{\mathop{\mathchar
58\relax}}\Omega\to\mathbb{R}$, we consider the elliptic problem consisting
into finding a function $v\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}^{N}\to\mathbb{R}$ satisfying of the following problem:

$\displaystyle\begin{cases}\mathscr{L}v+\varphi(v)\,v=f&\mbox{ in
}\;\Omega,\\\ v=0&\mbox{ on }\;\mathbb{R}^{N}\setminus\Omega.\end{cases}$
(3.1)

Heuristically, the problem (3.1) results from the evolution problem (1.1) by
integrating with respect to $t$ from $0$ to $T$. In a sense, the functions $v$
and $f$ correspond to $\int_{0}^{T}u(\cdot,t)\,\mathrm{d}t$ and
$u_{0}-u(\cdot,T)$, respectively. Semilinear problems of type (3.1) are
considered in the classical scenario in [BEB78, Gos71, Hes73, Sta21]) with the
operator $\mathscr{L}$ replaced with $-\Delta$. There, the difficulties with
the integrability of the term $\varphi(v)v$ were handled. In our case, we
consider the function $f\in L^{2}(\Omega)$, so that we expect more from the
solution of the problem such as $\varphi(v)\in L^{2}(\Omega)$. We need to
introduce the following notation

$\upchi(\tau)=\varphi(\tau)\tau.$

A function $v\in\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N})$ is said to be a weak
solution of problem (3.1) if $\upchi(v)\in L^{2}(\Omega)$ and

$\displaystyle\mathcal{E}(v,\psi)+(\upchi(v),\psi)=(f,\psi)\quad\text{for
all}\quad\psi\in\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N}).$ (3.2)

Next, we want to show that the above variational problem (3.2) is well-posed
in the sense of Hadamard. In other words, it possesses a unique solution which
continuously depends upon the data. Let us start with the following stability
lemma.

######  Lemma 3.1.

Let $f_{i}\in L^{2}(\Omega)$, $i=1,2$. Assume that
$v_{i}\in\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N})$ satisfies

$\displaystyle\mathcal{E}(v_{i},\psi)+(\upchi(v_{i}),\psi)=(f_{i},\psi)\quad\text{for
all}\quad\psi\in\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N}).$

Then for some constant $C=C(N,\Omega,\nu)>0$ only depending only $N,\Omega$
and $\nu$ such that

$\displaystyle\|v_{1}-v_{2}\|_{\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N})}\leq
C\|f_{1}-f_{2}\|_{(\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N}))^{*}}.$

###### Proof.

Combining both equation and testing with $\psi=v_{1}-v_{2}$ yields

$\displaystyle\mathcal{E}(v_{1}-v_{2},v_{1}-v_{2})+\big{(}\upchi(v_{1})-\upchi(v_{2}),v_{1}-v_{2}\big{)}_{L^{2}(\Omega)}=(f_{1}-f_{2},v_{1}-v_{2})_{L^{2}(\Omega)}.$

Observing that $\tau\mapsto\upchi(\tau)=\varphi(\tau)\tau$ is non-decreasing,
is equivalent to saying that

$\displaystyle(\upchi(\tau_{1})-\upchi(\tau_{2}))(\tau_{1}-\tau_{2})\geq
0\qquad\text{for all $\tau_{1},\tau_{2}\in\mathbb{R},$}$ (3.3)

the above relation implies

$\displaystyle\|v_{1}-v_{2}\|^{2}_{\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N})}\leq\|f_{1}-f_{2}\|_{(\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N}))^{*}}\|v_{1}-v_{2}\|_{\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N})}.$

The desired estimate follows from the Poincaré-Friedrichs inequality (2.2). ∎

The next result reminisces [Sta21, Lemma 1, Section 3.1] in the nonlocal
setting.

######  Theorem 3.2.

Let Assumption 1.1 be in force and let $f\in L^{2}(\Omega)$. Then the problem
(3.1) has a unique weak solution
$v\in\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N})$. Moreover, the following
estimates hold true:

  1. $(i)$

$\mathcal{E}(v,v)\leq C\,\|f\|^{2}_{L^{2}(\Omega)}$ where $C>0$ only depends
on $N$, $\Omega$, and $\nu$;

  2. $(ii)$

$\|\varphi(v)\,v\|_{L^{2}(\Omega)}\leq\|f\|_{L^{2}(\Omega)}$;

  3. $(iii)$

$\|\varphi(v)\|^{2}_{L^{2}(\Omega)}\leq\frac{1}{\delta^{2}}\|f\|^{2}_{L^{2}(\Omega)}+|\Omega|$,
with $\delta>0$ only depending on $\varphi$.

###### Proof.

Note that the uniqueness immediately follows from Lemma 3.1. We prove the
remaining results of Theorem 3.2 in several steps. Our proof follows that of
[Sta21, Lemma 1, Section 3.1].

Step 1: We are interested in establishing the well-posedness of problem (3.1)
using the Galerkin method which consists into projecting the latter on
suitable finite dimensional space. First of all, we mention that bounded
functions are dense in $V_{\nu}(\Omega|\mathbb{R}^{N})$ and hence in
$\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N})$. Thus, there is an orthonormal basis
$\\{\phi_{k}\\}$ of $\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N})$ whose elements
are bounded, i.e., $\phi_{k}\in L^{\infty}(\Omega)$.

We emphasize that the inner product in
$\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N})$ is defined as
$(\psi_{1},\psi_{2})_{\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N})}=\mathcal{E}(\psi_{1},\psi_{2})$
for $\psi_{1}$, $\psi_{2}\in\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N})$. Let
$\mathcal{V}_{k}$ be the subspace of $\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N})$
spanned by the basis functions $\\{\phi_{1},\ldots,\phi_{k}\\}$. For each
$k\in\mathbb{N}$, we claim the existence of a function
$v_{k}\in\mathcal{V}_{k}$ such that

$\displaystyle\mathcal{E}(v_{k},\psi)+(\upchi(v_{k}),\psi)=(f,\psi)\quad\text{for
all}\quad\psi\in\mathcal{V}_{k}.$ (3.4)

We prove this in two different ways. First, note that (3.4) is equivalent to
the minimization problem

$\displaystyle\mathcal{J}(v_{k})=\min_{w\in\mathcal{V}_{k}}\mathcal{J}(w)\quad\text{with}\quad\mathcal{J}(w)\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{2}\mathcal{E}(w,w)+\int_{\Omega}G(w)\mathrm{d}x+\int_{\Omega}fw\mathrm{d}x$

where we define the function
$G(w)=\int_{0}^{w}\chi(\tau)\mathrm{d}\tau=\int_{0}^{w}\varphi(\tau)\tau\mathrm{d}\tau$.
Note that $G$ is non-negative since $\varphi(\tau)\geq 0$ and that the mapping
$w\mapsto\mathcal{J}(w)$ is continuous on $\mathcal{V}_{k}$. Furthermore, with
the aid of the Poincaré-Friedrichs inequality (2.2) we find that
$\mathcal{J}(w)\to\infty$, as
$\|w\|_{\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N})}\to\infty$ and
$w\in\mathcal{V}_{k}$. Since $\dim\mathcal{V}_{k}<\infty$, the existence of a
minimizer $v_{k}\in\mathcal{V}_{k}$ of $\mathcal{J}$ springs from folklore
arguments.

Alternatively, as highlighted in [Sta21], we obtain the existence of $v_{k}$
using the Brouwer fixed-point theorem as follows. Let $w\in\mathcal{V}_{k}$,
necessarily $\varphi(w)$ is a bounded function since $\phi_{k}$’s are also
bounded. The Lax-Milgram lemma implies there is a unique function
$\widehat{w}\in\mathcal{V}_{k}$ such that

$\displaystyle\mathcal{E}(\widehat{w},\psi)+(\varphi(w)\widehat{w},\psi)=(f,\psi)\quad\text{for
all}\quad\psi\in\mathcal{V}_{k}.$

In particular, the Poincaré–Friedrichs inequality (2.2) yields

$\displaystyle\begin{aligned}
\mathcal{E}(\widehat{w},\widehat{w})+\int_{\Omega}\varphi(w)\widehat{w}^{2}\,\mathrm{d}x\leq&\|f\|_{L^{2}(\Omega)}\,\|\widehat{w}\|_{L^{2}(\Omega)}\leq
C\|f\|_{L^{2}(\Omega)}\,\|\widehat{w}\|_{\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N})}\end{aligned}$

Thus, letting $R=C\,\|f\|_{L^{2}(\Omega)}$, since $\varphi\geq 0$ we obtain
the following estimates

$\displaystyle\|\widehat{w}\|_{\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N})}\leq
R\quad\text{ and }\quad\int_{\Omega}\varphi(w)\widehat{w}^{2}\,\mathrm{d}x\leq
R^{2}.$ (3.5)

We let $\mathcal{B}_{R}=\big{\\{}w\in\mathcal{V}_{k}\mathrel{\mathop{\mathchar
58\relax}}\|w\|_{\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N})}\leq R\big{\\}}$, be
the closed ball in $\mathcal{V}_{k}$ of radius $R$ centered at the origin.
Clearly, (3.5) implies that the mapping $T\mathrel{\mathop{\mathchar
58\relax}}\mathcal{V}_{k}\to\mathcal{B}_{R}$ with $Tw=\widehat{w}$ is well
defined. It remains to prove that $T$ is a continuous mapping. Indeed, let
$\\{w_{n}\\}$ be a sequence in $\mathcal{V}_{k}$ with
$w_{n}=\lambda_{1,n}\phi_{1}+\cdots+\lambda_{k,n}\phi_{k}$ converging in
$\mathcal{V}_{k}$ to a function
$w=\lambda_{1}\phi_{1}+\cdots+\lambda_{k}\phi_{k}$, i.e.,
$\lambda_{\ell,n}\xrightarrow{n\to\infty}\lambda_{\ell}$, $\ell=1,2,\cdots,k$.
By continuity we have $\varphi(w_{n})\xrightarrow{n\to\infty}\varphi(w)$
almost everywhere. In addition, the convergence in $L^{2}(\Omega)$ also holds,
i.e., $\|\varphi(w_{n})-\varphi(w)\|_{L^{2}(\Omega)}\xrightarrow{n\to\infty}0$
since the continuity gives $\sup_{n\geq
0}\|\varphi(w_{n})\|_{L^{\infty}(\Omega)}<\infty$ because $\sup_{n\geq
0}\|w_{n}\|_{L^{\infty}(\Omega)}<\infty$. On the other side, in virtue of the
first estimate in (3.5), the sequence $\\{Tw_{n}\\}$ is bounded in finite
dimensional space $\mathcal{V}_{k}$ and thus converges in $\mathcal{V}_{k}$ up
to a subsequence to some $w_{*}\in\mathcal{V}_{k}$. Altogether, it follows
that, for all $\psi\in\mathcal{V}_{k}\subset L^{\infty}(\Omega)$

$\displaystyle(f,\psi)=\lim_{n\to\infty}\mathcal{E}(\widehat{w}_{n},\psi)+(\varphi(w_{n})\widehat{w}_{n},\psi)=\mathcal{E}(w_{*},\psi)+(\varphi(w)w_{*},\psi).$

The uniqueness of $\widehat{w}$ entails that $w_{*}=\widehat{w}=Tw$ and hence
the whole sequence $\\{Tw_{n}\\}$ converges in $Tw$ in $\mathcal{V}_{k}$,
which gives the continuity of $T$. Therefore, by the Brouwer fixed-point
theorem, $T$ has a fixed point $v_{k}\in\mathcal{V}_{k}$, i.e., $v_{k}=Tv_{k}$
which clearly satisfies (3.4) as announced.

To continue, we must show that a subsequence of $\\{v_{k}\\}$ converges in
$L^{2}(\Omega)$. To do this, we recall that $R=C\,\|f\|_{L^{2}(\Omega)}$ so
that from (3.5) we get the following estimates

$\displaystyle\|v_{k}\|_{\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N})}\leq
R\quad\text{and}\quad\int_{\Omega}\upchi(v_{k})\,v_{k}\,\mathrm{d}x\leq
R^{2}\qquad\text{ for all $k\in\mathbb{N}$}.$ (3.6)

Therefore, the sequence $\\{v_{k}\\}$ is clearly bounded in
$\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N})$. The compactness Theorem 2.2 yields
the existence of a subsequence, still denoted by $\\{v_{k}\\}$, converging
weakly in $\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N})$, strongly in
$L^{2}(\Omega)$ and almost everywhere in $\Omega$ to a function $v$.
Wherefore, due to the continuity of $\upchi$, we get

$\displaystyle\upchi(v_{k})\to\upchi(v)\quad\text{almost everywhere
in}\leavevmode\nobreak\ \leavevmode\nobreak\ \Omega.$ (3.7)

Step 2: Next, we prove that the functions $\\{\upchi(v_{k})\\}$ are uniformly
integrable. In view of the estimate (3.6), for each measurable set
$\Gamma\subset\Omega$ and each $\Lambda>0$, we let
$\Gamma_{\Lambda}^{k}=\\{x\in\Gamma\,\mathrel{\mathop{\mathchar
58\relax}}\,|v_{k}(x)|\geq\Lambda\\}$ so that

$\int_{\Gamma_{\Lambda}^{k}}|\upchi(v_{k})|\,\mathrm{d}x\leq\frac{1}{\Lambda}\int_{\Omega}\upchi(v_{k})\,v_{k}\,\mathrm{d}x\leq\frac{R^{2}}{\Lambda}.$

Since $\upchi$ is non-decreasing, putting
$\upgamma(\Lambda)=\Lambda\max\\{\varphi(-\Lambda),\varphi(\Lambda)\\}$, we
get

$\displaystyle|\upchi(\tau)|\leq\upgamma(\Lambda)\quad\text{for
all}\quad\tau\in[-\Lambda,\Lambda].$

Therefore, the following relation holds

$\int_{\Gamma\setminus\Gamma_{\Lambda}^{k}}|\upchi(v_{k})|\,\mathrm{d}x\leq\upgamma(\Lambda)\,|\Gamma|,$

where $|\Gamma|$ is the Lebesgue measure of the set $\Gamma$. These
inequalities imply that

$\int_{\Gamma}|\upchi(v_{k})|\,\mathrm{d}x\leq\frac{R^{2}}{\Lambda}+\upgamma(\Lambda)\,|\Gamma|.$

Thus, for an arbitrary $\varepsilon>0$, we take $\Lambda=2R^{2}/\varepsilon$
and $\updelta=\varepsilon/(2\upgamma(\Lambda))$. Therefore, we find that

$\sup_{k\geq 1}\int_{\Gamma}|\upchi(v_{k})|\,\mathrm{d}x<\varepsilon$

for an arbitrary measurable set $\Gamma\subset\Omega$ such that
$|\Gamma|<\delta$. This, is precisely the uniform integrability of
$\upchi(v_{k})$. This fact together with (3.7) and the Vitali convergence
theorem (see, e.g., [Fog20, Theorem A.19]) enable us to conclude that
$\upchi(v)\in L^{1}(\Omega)$ and $\upchi(v_{k})\to\upchi(v)$ in
$L^{1}(\Omega)$ as $k\to\infty$. Now passing to the limit in (3.4) as
$k\to\infty$ we find that $v$ satisfies (3.2), which along with Lemma 3.1,
means that $v$ is a unique weak solution of problem (3.1).

Step 3: We prove the estimates in $(i),(ii)$ and $(iii)$. The estimate (i)
follows from the first inequality in (3.6) since the weak convergence of
$(v_{k})_{k}$ implies

$\mathcal{E}(v,v)\leq\liminf\limits_{k\to\infty}\mathcal{E}(v_{k},v_{k})\leq
C\|f\|^{2}_{L^{2}(\Omega)}.$

Next, let us consider the truncation
$\bar{v}_{\ell}=\max(-\ell,\min(\ell,v)),$ $\ell\geq 1$. Then
$\bar{v}_{\ell}\in\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N})$ since
$(v(x)-v(y))(\bar{v}_{\ell}(x)-\bar{v}_{\ell}(y)\geq|\bar{v}_{\ell}(x)-\bar{v}_{\ell}(y)|^{2}$.
The latter inequality and (3.3) imply
$\mathcal{E}(v,\upchi(\bar{v}_{\ell}))\geq\mathcal{E}(\bar{v}_{\ell},\upchi(\bar{v}_{\ell}))\geq
0$. Moreover, $|\upchi(\bar{v}_{\ell})(x)-\upchi(\bar{v}_{\ell})(y)|\leq
c|\bar{v}_{\ell}(x)-\bar{v}_{\ell}(y)|$ since $\upchi$ is Lipschitz on
$[-\ell,\ell]$ and $|\bar{v}_{\ell}|\leq\ell$. Thus,
$\upchi(\bar{v}_{\ell})\in\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N})$. Noticing
that, $\upchi(v)\upchi(\bar{v}_{\ell})\geq\upchi^{2}(\bar{v}_{\ell}),$ taking
$\psi=\upchi(\bar{v}_{\ell})$ in (3.2) yields

$\displaystyle\|\upchi(\bar{v}_{\ell})\|^{2}_{L^{2}(\Omega)}$
$\displaystyle\leq\mathcal{E}(v,\upchi(\bar{v}_{\ell}))+(\upchi(v),\upchi(\bar{v}_{\ell}))\leq\|f\|_{L^{2}(\Omega)}\|\upchi(\bar{v}_{\ell})\|_{L^{2}(\Omega)}.$

Thus, since $\\{\upchi^{2}(\bar{v}_{\ell})\\}\to\upchi^{2}(v)$ a.e. in
$\Omega$, Fatou’s lemma implies the second estimate as follows

$\displaystyle\|\upchi(v)\|_{L^{2}(\Omega)}\leq\liminf_{\ell\to\infty}\|\upchi(\bar{v}_{\ell})\|_{L^{2}(\Omega)}\leq\|f\|_{L^{2}(\Omega)}.$

Finally, by continuity of $\varphi$, there exists $\delta>0$ such that
$\varphi^{2}(\tau)\leq 1$ for all $\tau\in[-\delta,\delta]$. Hence, letting
$\Gamma_{\delta}=\\{x\in\Omega\;\mathrel{\mathop{\mathchar
58\relax}}\;|v(x)|\geq\delta\\}$, the second estimate implies the third one as
follows

$\displaystyle\int_{\Omega}\varphi^{2}(v)\,\mathrm{d}x$
$\displaystyle=\int_{\Gamma_{\delta}}\varphi^{2}(v)\,\mathrm{d}x+\int_{\Omega\setminus\Gamma_{\delta}}\varphi^{2}(v)\,\mathrm{d}x\leq\frac{1}{\delta^{2}}\int_{\Omega}\varphi^{2}(v)\,v^{2}\,\mathrm{d}x+\int_{\Omega\setminus\Gamma_{\delta}}\varphi^{2}(v)\,\mathrm{d}x\leq\frac{1}{\delta^{2}}\|f\|^{2}_{L^{2}(\Omega)}+|\Omega|.$

∎

Next, we define the mapping $V\mathrel{\mathop{\mathchar
58\relax}}L^{2}(\Omega)\to\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N})$ such that,
for $f\in L^{2}(\Omega)$,

$v=V(f)$ is the unique weak solution of problem (3.1). (3.8)

We derive in the lemma below, some convergence results for the sequence
$\\{V(f_{k})\\}$ which are decisive for the application the Tychonoff fixed-
point Theorem 2.5.

######  Lemma 3.3.

Assume that $f_{k}\rightharpoonup f$ weakly $L^{2}(\Omega)$ and let
$v_{k}=V(f_{k})$ and $v=V(f)$. Then it holds that $v_{k}\to v$ strongly in
$\mathbb{X}_{\nu}(\Omega|{\mathbb{R}}^{N})$ and
$\varphi(v_{k})\rightharpoonup\varphi(v)$ weakly in $L^{2}(\Omega)$.

###### Proof.

Let us identify $f_{k}-f$ in $(\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N}))^{*}$
with the linear form

$\displaystyle w\mapsto\int_{\Omega}(f_{k}(x)-f(x))w(x)\mathrm{d}x.$

By the reflexivity of $\mathbb{X}_{\nu}(\Omega|{\mathbb{R}}^{N})$, there
exists $w_{k}\in X_{\nu}(\Omega|{\mathbb{R}}^{N})$ (cf. [Jam64, Theorem 2],
[KK01, Chapter 6] or [Che13, page 60]) such that
$\|w_{k}\|_{\mathbb{X}_{\nu}(\Omega|{\mathbb{R}}^{N})}\leq 1$ and

$\displaystyle\|f_{k}-f\|_{\mathbb{X}_{\nu}((\Omega|\mathbb{R}^{N}))^{*}}=\int_{\Omega}(f_{k}(x)-f(x))w_{k}(x)\mathrm{d}x.$

According to the compactness Theorem 2.2 we may assume that $\\{w_{k}\\}$
strongly converges to some $w$ in $L^{2}(\Omega)$. Therefore, the weakly
convergence of $\\{f_{k}\\}$ implies that

$\displaystyle\|f_{k}-f\|_{\mathbb{X}_{\nu}((\Omega|\mathbb{R}^{N}))^{*}}=\int_{\Omega}(f_{k}(x)-f(x))w_{k}(x)\mathrm{d}x\xrightarrow{k\to\infty}0.$

The convergence in $\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N})$ follows
immediately from Lemma 3.1 since

$\displaystyle\|v_{k}-v\|_{\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N})}\leq
C\|f_{k}-f\|_{(\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N}))^{*}}\xrightarrow{k\to\infty}0.$

On the other hand, we also have the strong convergence of $\\{v_{k}\\}$ in
$L^{2}(\Omega)$ and the continuity of $\varphi$ imply that
$\\{\varphi(v_{k})\\}$ converges almost everywhere to $\varphi(v)$ up a
subsequence. Furthermore, since $\\{f_{k}\\}$ is bounded, as in the proof of
Lemma 3.1, one easily gets that

$\displaystyle\|\varphi(v_{k})\|_{L^{2}(\Omega)}\leq C\quad\text{for all
$k\geq 1$,}$

for a constant $C>0$ independent on $k$. Thus, $\\{\varphi(v_{k})\\}$ has a
further subsequence weakly converging in $L^{2}(\Omega)$. The Banach-Saks
Theorem, see [MR12, Appendix A] or [Pon16, Proposition 10.8], infers the
existence of a further subsequence whose Césaro mean converges strongly in
$L^{2}(\Omega)$ and almost everywhere in $\Omega$ to the same limit.
Necessarily, since $\\{\varphi(v_{k})\\}$ converges almost everywhere to
$\varphi(v)$, the entire sequence $\\{\varphi(v_{k})\\}$ weakly converges in
$L^{2}(\Omega)$ to $\varphi(v)$. ∎

###  3.2. Nonlocal parabolic problem

We consider the following parabolic problem:

$\displaystyle\begin{cases}\partial_{t}u+\mathscr{L}u+\zeta u=0&\mbox{ in
}\;\Omega_{T},\\\ u=0&\mbox{ in }\;\Sigma,\\\ u(\cdot,0)=u_{0},&\mbox{ in
}\;\Omega,\end{cases}$ (3.9)

where $u_{0},\zeta\in L^{2}(\Omega)$ with $\zeta\geq 0$. We also assume
$\nu\not\in L^{1}(\mathbb{R}^{N})$ so that by Theorem 2.2, the embedding
$\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N})\hookrightarrow L^{2}(\Omega)$ is
compact. Therefore, by the standard Galerkin superposition method (see for
instance [Fog20, Section 4.6]), a weak solution $u$ of the problem (3.9) can
be easily obtained in
$L^{2}\big{(}0,T;\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N})\cap
L^{2}(\Omega,\zeta)\big{)}$. Here $L^{2}(\Omega,\zeta)$ is the Hilbert space
with the norm

$\displaystyle\|u\|_{L^{2}(\Omega,\zeta)}^{2}=\int_{\Omega}|u(x)|^{2}\,\zeta(x)\mathrm{d}x.$

We omit the proof as well as various justifications (see also [Eva10, GGZ74]).
Another possibility, is to observe that (see [Kat13]) there exists a unique
semigroup with generator $A$ on $L^{2}(\Omega)$ associated to the closed
bilinear form
$a(u,v)=(u,v)_{\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N})}+(u,v)_{L^{2}(\Omega,\zeta)}$,
with $u,v\in\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N})\cap L^{2}(\Omega,\zeta)$,
such that $a(u,v)=\langle Au,v\rangle$. Thus $u(x,t)=e^{-tA}u_{0}(x),\,\,0\leq
t\leq T$, is the unique weak solution to (3.9). The weak solution of problem
(3.9) satisfies the energy estimate:

$\displaystyle\frac{1}{2}\,\|u(\cdot,t)\|^{2}_{L^{2}(\Omega)}+\int_{0}^{t}\mathcal{E}(u,u)\,\mathrm{d}\tau+\int_{0}^{t}\int_{\Omega}\zeta\,u^{2}\,\mathrm{d}x\,\mathrm{d}\tau\leq\frac{1}{2}\,\|u_{0}\|^{2}_{L^{2}(\Omega)}$
(3.10)

for all $t\in[0,T)$. Besides that, $\partial_{t}u$ belongs to the space
$L^{2}\big{(}0,T;(\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N})\cap
L^{2}(\Omega,\zeta))^{*}\big{)}$, where
$(\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N})\cap L^{2}(\Omega,\zeta))^{*}$ is the
dual space of $\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N})\cap
L^{2}(\Omega,\zeta)$. According to Proposition 2.3 (see (2.5)) we find that
$u\in C(0,T;L^{2}(\Omega))$. Thus, the function $u_{T}=u(\cdot,T)$ is well
defined and belongs to $L^{2}(\Omega)$. Moreover, the estimate (3.10) holds in
particular for $t=T$.

For each $\zeta\in L^{2}(\Omega)$, $\zeta\geq 0$, define the mapping
$\mathscr{U}\mathrel{\mathop{\mathchar
58\relax}}\zeta\mapsto\mathscr{U}(\zeta)$ where $\mathscr{U}(\zeta)\in
L^{2}\big{(}0,T;\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N})\cap
L^{2}(\Omega,\zeta)\big{)}$ is the unique weak solution of problem (3.9).

We want to study the continuity of the operators $\mathscr{U}$ and
$\mathscr{U}_{T}$ on $L^{2}(\Omega)$, with
$\mathscr{U}_{T}(\zeta)(\cdot)=\mathscr{U}(\zeta)(\cdot,T)$.

######  Lemma 3.4.

Let $u_{0}\in L^{2}(\Omega)$ and $\\{\zeta_{k}\\}$ be a sequence of non-
negative functions converging weakly in $L^{2}(\Omega)$ to a function $\zeta$.
Then $\mathscr{U}_{T}(\zeta_{k})\rightharpoonup\mathscr{U}_{T}(\zeta)$ weakly
in $L^{2}(\Omega)$ as $k\to\infty$.

###### Proof.

For brevity, we denote $u_{k}=\mathscr{U}(\zeta_{k})$ and
$u=\mathscr{U}(\zeta)$. It follows from (3.10), for all $k\in\mathbb{N}$,

$\displaystyle\frac{1}{2}\,\|u_{k}(\cdot,T)\|^{2}_{L^{2}(\Omega)}+\int_{0}^{T}\mathcal{E}(u_{k},u_{k})\,\mathrm{d}t+\int_{0}^{T}\int_{\Omega}\zeta_{k}\,u_{k}^{2}\,\mathrm{d}x\,\mathrm{d}t\leq\frac{1}{2}\,\|u_{0}\|^{2}_{L^{2}(\Omega)}.$
(3.11)

Next, consider $\psi\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}^{N}\times(0,T)\to\mathbb{R}$ be an arbitrary smooth
function such that $\psi=0$ in $\mathbb{R}^{N}\setminus\Omega\times(0,T)$.
Define $w_{k}(x)=\int_{0}^{T}u_{k}(x,t)\psi(x,t)\mathrm{d}t$ assuming
$\psi(\cdot,t)\in C_{c}^{\infty}(\Omega)$ there holds

$|\psi(x,t)-\psi(y,t)|\leq\|\psi(\cdot,t)\|_{W^{1,\infty}(\mathbb{R}^{N})}(1\land|x-y|).$

Using this inequality yields

$\displaystyle|w_{k}(x)-w_{k}(y)|^{2}$
$\displaystyle=\Big{|}\int_{0}^{T}(u_{k}(x,t)-u_{k}(y,t))\psi(x,t)\
\mathrm{d}t+\int_{0}^{T}(\psi(x,t)-\psi(y,t))u_{k}(y,t)\mathrm{d}t\Big{|}^{2}$
$\displaystyle\leq
2T\int_{0}^{T}|u_{k}(x,t)-u_{k}(y,t)|^{2}|\psi(x,t)|^{2}\mathrm{d}t+2T\int_{0}^{T}|\psi(x,t)-\psi(x,t)|^{2}|u_{k}(y,t)|^{2}\mathrm{d}t$
$\displaystyle\leq
2T\sup_{t\in[0,T]}\|\psi(\cdot,t)\|^{2}_{W^{1,\infty}(\mathbb{R}^{N})}\int_{0}^{T}|u_{k}(x,t)-u_{k}(y,t)|^{2}+(1\land|x-y|^{2})|u_{k}(y,t)|^{2}\mathrm{d}t.$

Integrating both side over $\Omega\times\mathbb{R}^{N}$ (see Remark 2.1) with
respect to the measure $\nu(x-y)\mathrm{d}x\mathrm{d}y$ gives

$\displaystyle\mathcal{E}(w_{k},w_{k})$ $\displaystyle\leq
2T\sup_{t\in[0,T]}\|\psi(\cdot,t)\|^{2}_{W^{1,\infty}(\mathbb{R}^{N})}\int_{0}^{T}\mathcal{E}(u_{k},u_{k})\mathrm{d}t+\int_{\mathbb{R}^{N}}(1\land|h|^{2})\nu(h)\mathrm{d}h\int_{0}^{T}\int_{\Omega}|u_{k}(,t)|^{2}\mathrm{d}x\mathrm{d}t.$

Altogether, with the Poincaré inequality (2.2) and the inequality (3.11) yield

$\displaystyle\frac{1}{2}\,\|u_{k}(\cdot,T)\|^{2}_{L^{2}(\Omega)}+\int_{0}^{T}\mathcal{E}(u_{k},u_{k})\,\mathrm{d}t$
$\displaystyle\leq\frac{1}{2}\,\|u_{0}\|^{2}_{L^{2}(\Omega)}\quad\text{and}\quad\mathcal{E}(w_{k},w_{k})\leq
C_{\psi}\|u_{0}\|^{2}_{L^{2}(\Omega)}$

Therefore, $\\{w_{k}\\}$ is bounded in
$\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N})$. By the compactness Theorem 2.2 we
can assume that $\\{w_{k}\\}$ strongly converge to some $w\in L^{2}(\Omega)$.
On the other hand, $\\{u_{k}\\}$ and $\\{u_{k}(\cdot,T)\\}$ are bounded in
$L^{2}\big{(}0,T;\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N}))$ and $L^{2}(\Omega)$
respectively and hence can be assumed to weakly converge to some $u\in
L^{2}\big{(}0,T;\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N}))$ and $h\in
L^{2}(\Omega)$ respectively. In particular, the strong convergence implies
that, for every $\phi\in C_{c}^{\infty}(\Omega)$,

$\displaystyle\int_{\Omega}w(x)\phi(x)\mathrm{d}x=\lim_{k\to\infty}\int_{\Omega}w_{k}(x)\phi(x)\mathrm{d}x=\lim_{k\to\infty}\int_{\Omega}\phi(x)\int_{0}^{T}u_{k}(x,t)\mathrm{d}t\mathrm{d}x=\int_{\Omega}\phi(x)\int_{0}^{T}u(x,t)\mathrm{d}t\mathrm{d}x.$

It turns out that $w=\int_{0}^{T}u(\cdot,t)\mathrm{d}t$ a.e. on $\Omega$. By
the same token, one gets $h=u(\cdot,T)$ a.e. on $\Omega$. Once again, the
strong convergence of $\\{w_{k}\\}$ and the weak of convergence of
$\\{\zeta_{k}\\}$ in $L^{2}(\Omega)$ yield

$\displaystyle\int_{0}^{T}\int_{\Omega}\zeta_{k}u_{k}\psi\,\mathrm{d}x\,\mathrm{d}t\xrightarrow{k\to\infty}\int_{0}^{T}\int_{\Omega}\zeta
u\psi\,\mathrm{d}x\,\mathrm{d}t.$ (3.12)

For each $k\geq 1$, by definition of $u_{k}=\mathscr{U}(\zeta_{k})$, we get

$\displaystyle\int_{0}^{T}\int_{\Omega}u_{k}\partial_{t}\psi\,\mathrm{d}x\,\mathrm{d}t+\int_{0}^{T}\mathcal{E}(u_{k},\psi)\,\mathrm{d}t+\int_{0}^{T}\int_{\Omega}\zeta_{k}u_{k}\,\psi\,\mathrm{d}x\,\mathrm{d}t\,\mathrm{d}t=-\int_{\Omega}(u_{k}(\cdot,T)\psi(\cdot,T)-u_{0}\psi(\cdot,0))\,\mathrm{d}x.$

By choosing in particular the test function $\psi(\cdot,T)=0$, letting
$k\to\infty$ yields $w(\cdot,0)=u_{0}$ and

$\displaystyle\int_{0}^{T}\int_{\Omega}\partial_{t}w\psi\,\mathrm{d}x+\int_{0}^{T}\mathcal{E}(w,\psi)\,\mathrm{d}t+\int_{0}^{T}\int_{\Omega}\zeta
w\psi\mathrm{d}x\mathrm{d}t=0\quad\text{for all $\psi\in
C^{1}_{c}(0,T;C_{c}^{\infty}(\Omega))$}.$

This means that $u$ is a weak solution to (3.9) and by uniqueness,
$u=\mathscr{U}(\zeta)$. The uniqueness of $u$ implies that the whole sequence
$\\{u_{k}\\}$ weakly converges to $u$ in
$L^{2}\big{(}0,T;\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N}))$. Therefore, since
$u_{k}=\mathscr{U}(\zeta_{k})$ and $u=\mathscr{U}(\zeta)$ are weak solutions,
using (3.12) and the weak convergence, for $\psi\in L^{2}(\Omega)$ gives

$\displaystyle\int_{\Omega}(\mathscr{U}_{T}(\zeta_{k})-\mathscr{U}_{T}(\zeta))(x)\psi(x)\mathrm{d}x=\int_{0}^{T}\mathcal{E}(u_{k}-u,\psi)\,\mathrm{d}t+\int_{0}^{T}\int_{\Omega}(u_{k}\zeta_{k}-u\zeta)(x,t)\psi(x)\mathrm{d}x\mathrm{d}t\xrightarrow{k\to\infty}0.$

That is, the whole sequence $\\{\mathscr{U}_{T}(\zeta_{k})\\}$ weakly
converges to $U_{T}$ in $L^{2}(\Omega)$ which is the sought result. ∎

##  4\. Weak solvability and uniqueness of the solution

Armed with the above auxiliaries results, let us turn our attention to the
proof of the weak solvability of problem (1.1). In order to apply the
Tychonoff fixed-point Theorem 2.5, we take $X=L^{2}(\Omega)$, $G=\\{w\in
L^{2}(\Omega)\,\mathrel{\mathop{\mathchar
58\relax}}\;\|w\|_{L^{2}(\Omega)}\leq\|u_{0}\|_{L^{2}(\Omega)}\\}$ which is
clearly closed, convex and bounded. The next result provides the existence of
a weak solution to the problem (1.1).

######  Theorem 4.1.

Let $u_{0}\in L^{2}(\Omega)$ and $T>0$. Let the mapping
$\pi\mathrel{\mathop{\mathchar 58\relax}}G\to G$ with
$\pi(w)=\mathscr{U}_{T}(\varphi(v))$, where $v=V(u_{0}-w)$ (defined as in
(3.8)) is the unique weak solution to (3.1) with $f=u_{0}-w$. Then $\pi$ has a
fixed point $u_{T}$ that is

$\displaystyle
u_{T}=\pi(u_{T})=\mathscr{U}_{T}\big{(}\varphi(v)\big{)}\quad\text{with
$v=V(u_{0}-u_{T})$.}$ Moreover, $\displaystyle v=\int_{0}^{T}u\mathrm{d}t$ and
$\displaystyle u=\mathscr{U}\big{(}\varphi(v)\big{)}\quad\text{is a weak
solution of the problem \eqref{eq:main-equation}. }$

###### Proof.

For $v=V(u_{0}-w)$ with $w\in G$, we know from Theorem 3.2 that $\varphi(v)\in
L^{2}(\Omega)$. Thus for $\zeta=\varphi(v)\geq 0$, the function
$\mathscr{U}(\zeta)$ satisfies (3.10) which implies that
$\|\mathscr{U}_{T}(\zeta)\|_{L^{2}(\Omega)}\leq\|u_{0}\|_{L^{2}(\Omega)}$. In
particular, $\|\pi(w)\|_{L^{2}(\Omega)}\leq\|u_{0}\|_{L^{2}(\Omega)}$ for all
$w\in G$ and thus, $\pi(G)\subset G$. It remains to prove the weak sequential
continuity of $\pi$. Let $\\{w_{k}\\}$ be an arbitrary sequence in $G$ that
converges to $w\in G$ weakly in $L^{2}(\Omega)$. We need to prove that
$\pi(w_{k})\rightharpoonup\pi(w)$ weakly in $L^{2}(\Omega)$ as $k\to\infty$.
In virtue of Lemma 3.3, $v_{k}=V(u_{0}-w_{k})\to v=V(u_{0}-w)$ strongly in
$\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N})$ and
$\varphi(v_{k})\rightharpoonup\varphi(v)$ weakly in $L^{2}(\Omega)$ as
$k\to\infty$, where $v_{k}=V(u_{0}-w_{k})$ and $v=V(u_{0}-w)$. In turn, Lemma
3.4 implies that
$\pi(w_{k})=\mathscr{U}_{T}(\varphi(v_{k}))\rightharpoonup\mathscr{U}_{T}(\varphi(v))=\pi(w)$
weakly in $L^{2}(\Omega)$ as $k\to\infty$. Thus, according to Theorem 2.5,
$\pi$ has a fixed point $u_{T}=\pi(u_{T})$. Next, knowing that $u_{T}$ is a
fixed point of the mapping $\pi$, we show that $u=\mathscr{U}(\varphi(v))$,
with $v=V(u_{0}-u_{T})$, is a weak solution to the problem (1.1). Indeed,
recall that $u=\mathscr{U}(\varphi(v))$ is the unique weak solution to the
problem (3.9) with $\zeta=\varphi(v)$, i.e., $u(\cdot,0)=u_{0}$ and for all
$\psi\in C^{1}_{c}(0,T;\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N}))$,

$\displaystyle\int_{0}^{T}\int_{\Omega}\partial_{t}u\,\psi\,\mathrm{d}x\,\mathrm{d}t+\int_{0}^{T}\mathcal{E}(u,\psi)\,\mathrm{d}t+\int_{0}^{T}\int_{\Omega}\varphi(v)\,u\,\psi\mathrm{d}x\,\mathrm{d}t=0.$
(4.1)

Integrating by parts, for $\psi\in\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N})$
(time independent) we get

$\displaystyle\mathcal{E}\Big{(}\int_{0}^{T}u\mathrm{d}t,\psi\Big{)}+\int_{\Omega}\varphi(v)\psi\,\int_{0}^{T}u\,\mathrm{d}t\,\mathrm{d}x=\int_{\Omega}[u_{0}-u_{T}]\psi\,\mathrm{d}x.$

Thus, according to Theorem 3.2, $v=\int_{0}^{T}u(x,t)\mathrm{d}t$ is the
unique weak solution to the elliptic problem

$\displaystyle\mathscr{L}v+\varphi(v)\,v=u_{0}-u_{T}\,\,\mbox{ in
}\;\,\Omega\quad\text{and}\quad v=0\,\,\mbox{ on
}\;\,\mathbb{R}^{N}\setminus\Omega.$ (4.2)

We have shown that $v=V(u_{0}-u_{T})=\int_{0}^{T}u\mathrm{d}t.$ Therefore, we
obtain

$\displaystyle
u=\mathscr{U}\Big{(}\varphi\Big{(}\int_{0}^{T}u\,\mathrm{d}t\Big{)}\Big{)}$
(4.3)

which, according to the relation (4.1), implies that $u$ is a weak solution to
the problem (1.1). ∎

The main result of the paper is the following theorem.

######  Theorem 4.2.

Let $u_{0}\in L^{2}(\Omega)$, $T>0$ and $\varphi$ satisfies Assumption 1.1.
The problem (1.1) has a weak solution $u\in L^{\infty}(0,T;L^{2}(\Omega))\cap
L^{2}(0,T;\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N}))$ such that

$\displaystyle\varphi(v)\in L^{2}(\Omega),\quad\varphi(v)\,v\in
L^{2}(\Omega),\quad\varphi(v)\,u^{2}\in L^{1}(\Omega_{T}),\leavevmode\nobreak\
\text{and}\quad u\in C(0,T;L^{2}(\Omega)),$

where $\displaystyle v=\int_{0}^{T}u\,\mathrm{d}t$. Moreover, the following
estimates hold true

$\displaystyle\frac{1}{2}\,\|u\|^{2}_{L^{\infty}(0,T;L^{2}(\Omega))}+\|u\|^{2}_{L^{2}(0,T;\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N}))}+\int_{0}^{T}\int_{\Omega}\varphi(v)\,u^{2}\,\mathrm{d}x\,\mathrm{d}t\leq\frac{1}{2}\,\|u_{0}\|^{2}_{L^{2}(\Omega)},$
$\displaystyle\|\partial_{t}u\|^{2}_{L^{2}(0,T;(\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N})\cap
L^{2}(\Omega,\varphi(v))^{*})}\leq\frac{1}{2}\,\|u_{0}\|^{2}_{L^{2}(\Omega)}.$

###### Proof.

The existence of a weak solution to the problem (1.1) follows immediately from
Theorem 4.1. Furthermore, mimicking the estimate (3.10) yields

$\displaystyle\frac{1}{2}\,\|u\|^{2}_{L^{\infty}(0,T;L^{2}(\Omega))}+\|u\|^{2}_{L^{2}(0,T;\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N}))}+\int_{0}^{T}\int_{\Omega}\varphi(v)\,u^{2}\,\mathrm{d}x\,\mathrm{d}t\leq\frac{1}{2}\,\|u_{0}\|^{2}_{L^{2}(\Omega)}.$

Now, for each $\psi\in L^{2}(0,T;\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N})\cap
L^{2}(\Omega,\varphi(v))$, by definition of $u$, we have

$\displaystyle\left|\int_{0}^{T}\langle\partial_{t}u,\psi\rangle\mathrm{d}t\right|^{2}=\left|-\int_{0}^{T}\mathcal{E}(u,\psi)\,\mathrm{d}t-\int_{0}^{T}\int_{\Omega}\varphi(v)\,u\psi\,\mathrm{d}x\,\mathrm{d}t\right|^{2}$
$\displaystyle\leq\left(\int_{0}^{T}\mathcal{E}(u,u)\mathrm{d}t\right)\left(\int_{0}^{T}\mathcal{E}(\psi,\psi)\mathrm{d}t\right)+\left(\int_{0}^{T}\int_{\Omega}\varphi(v)\,u^{2}\mathrm{d}x\mathrm{d}t\right)\left(\int_{0}^{T}\int_{\Omega}\varphi(v)\psi^{2}\,\mathrm{d}x\mathrm{d}t\right)$
$\displaystyle\leq\frac{1}{2}\|u_{0}\|^{2}_{L^{2}(\Omega)}\left(\int_{0}^{T}\mathcal{E}(\psi,\psi)\,\mathrm{d}t+\int_{0}^{T}\int_{\Omega}\varphi(v)\psi^{2}\,\mathrm{d}x\mathrm{d}t\right).$

This implies that

$\displaystyle\|\partial_{t}u\|^{2}_{L^{2}(0,T;(\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N})\cap
L^{2}(\Omega,\varphi(v))^{*})}\leq\frac{1}{2}\,\|u_{0}\|^{2}_{L^{2}(\Omega)}.$

Therefore, we have $u\in
L^{2}\big{(}0,T;\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N})\cap
L^{2}(\Omega,\zeta)\big{)}$ and $\partial_{t}u\in
L^{2}\big{(}0,T;(\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N})\cap
L^{2}(\Omega,\zeta))^{*}\big{)}$ with
$\zeta=\varphi\Big{(}\int_{0}^{T}u\mathrm{d}t\Big{)}$ which implies that
$\varphi(v)u^{2}\in L^{1}(\Omega_{T})$. On the one hand, by definition of
$u=\mathscr{U}(\varphi(v))$ it follows that $u\in C(0,T;L^{2}(\Omega))$. On
the other hand, we know that $v=\int_{0}^{T}u\,\mathrm{d}t$ is the unique weak
solution to the problem (4.2) and hence from Theorem 3.2 we have
$\varphi(v),\varphi(v)v\in L^{2}(\Omega)$. This ends the proof. ∎

Next, we prove that problem (1.1) has a unique solution, provided that the
initial condition $u_{0}$ is bounded. Before, we need to establish the
following maximum principle result.

######  Lemma 4.3.

Let $u=u(x,t)$ be a weak solution of the problem (1.1), i.e., satisfies (2.9)
with $u_{0}\in L^{2}(\Omega)\cap L^{\infty}(\Omega)$ then
$\|u\|_{L^{\infty}(0,T;L^{\infty}(\Omega))}\leq\|u_{0}\|_{L^{\infty}(\Omega)}$
on $\Omega_{T},$ i.e., $|u|\leq\|u_{0}\|_{L^{\infty}(\Omega)}$ a.e. on
$\Omega_{T}.$

###### Proof.

Set $\Lambda=\|u_{0}\|_{L^{\infty}(\Omega)}$ and consider the convex function
$F\mathrel{\mathop{\mathchar 58\relax}}\mathbb{R}\to[0,\infty)$ defined by

$\displaystyle
F(\tau)=\begin{cases}\big{(}\tau+\Lambda\big{)}^{2}&\text{if}\leavevmode\nobreak\
\leavevmode\nobreak\ \tau<-\Lambda\\\ 0&\text{if}\leavevmode\nobreak\
\leavevmode\nobreak\ |\tau|\leq\Lambda\\\
\big{(}\tau-\Lambda\big{)}^{2}&\text{if}\leavevmode\nobreak\
\leavevmode\nobreak\ \tau>\Lambda.\end{cases}$

So that, $F(\tau)=0$ if and only if $|\tau|\leq\Lambda$, in particular
$F(u_{0})=0$ a.e. on $\Omega$. By convexity, $F^{\prime}$ is non-decreasing,
i.e., $(F^{\prime}(\tau_{1})-F^{\prime}(\tau_{2}))(\tau_{1}-\tau_{2})\geq 0$
for all $\tau_{1},\tau_{2}\in\mathbb{R}$ in particular, since
$F^{\prime}(0)=0$, we have $F^{\prime}(\tau_{1})\tau_{1}\geq 0$ for all
$\tau_{1}\in\mathbb{R}$. Furthermore,
$F^{\prime}(u(\cdot,t))\in\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N})$ because
$u(\cdot,t)\in\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N})$ and one can check that
$F^{\prime}$ is Lipschitz since $F^{\prime\prime}$ is bounded. Therefore,
testing the equation (2.9) against $\zeta=F^{\prime}(u)$ gives

$\displaystyle\frac{d}{\mathrm{d}t}\int_{\Omega}F(u(x,t))\mathrm{d}x=-\mathcal{E}(u,F^{\prime}(u))-\int_{\Omega}\varphi(v)(x)F^{\prime}(u(x,t))u(x,t)\mathrm{d}x\leq
0.$

Since $F(u_{0})=0$ almost everywhere on $\Omega$, integrating the inequality
gives

$\displaystyle\int_{\Omega}F(u(x,t))\mathrm{d}x\leq 0\quad\text{for all }\quad
0\leq t\leq T.$

Thus, $F(u(x,t))=0$ a.e. on $\Omega_{T}$, and hence $|u(x,t)|\leq\Lambda$ a.e.
on $\Omega_{T}$. ∎

######  Theorem 4.4.

Assume that $\varphi$ satisfies Assumption 1.1., $u_{0}\in L^{2}(\Omega)\cap
L^{\infty}(\Omega)$ and that for $\Lambda=\|u_{0}\|_{L^{\infty}(\Omega)}$ we
have $|\varphi^{\prime}(\tau)|\leq\kappa$ for $\tau\in[-\Lambda T,\Lambda T]$
for some constant $\kappa>0$. Then the weak solution of problem (1.1) is
unique provided that $\kappa\Lambda T^{2}<1$.

###### Proof.

Suppose that problem (1.1) has two weak solutions $u_{1}$ and $u_{2}$, and put
$\displaystyle v_{i}(x)=\int_{0}^{T}u_{i}(x,t)\,\mathrm{d}t$, $i=1,2$. Then
$u=u_{1}-u_{2}$ is a weak solution to

$\begin{cases}\partial_{t}u+\mathscr{L}u+\varphi(v_{1})\,u_{1}-\varphi(v_{2})\,u_{2}=0&\mbox{
in }\;\Omega_{T},\\\ u=0&\mbox{ in }\;\Sigma,\\\ u(\cdot,0)=0&\mbox{ in
}\;\Omega.\end{cases}$

The maximum principle in Lemma 4.3, implies that $|u_{i}|\leq\Lambda$ a.e. in
$\Omega_{T}$ and hence $|v_{i}|\leq\Lambda T$, $i=1,2$, a.e. in $\Omega$.
Testing the above equation with $u$ leads to the following equality:

$\frac{1}{2}\,\frac{\mathrm{d}}{\mathrm{d}t}\|u(\cdot,t)\|^{2}_{L^{2}(\Omega)}+\mathcal{E}(u,u)+\int_{\Omega}\varphi(v_{1})\,u^{2}\,\mathrm{d}x+\int_{\Omega}\big{(}\varphi(v_{1})-\varphi(v_{2})\big{)}\,u_{2}\,u\,\mathrm{d}x=0$

which implies that

$\frac{1}{2}\,\|u(\cdot,t)\|^{2}_{L^{2}(\Omega)}+\int_{0}^{t}\mathcal{E}(u,u)\mathrm{d}\tau\leq\kappa\Lambda\int_{0}^{t}\int_{\Omega}|v(x)|\,|u(x,\tau)|\,\mathrm{d}x\mathrm{d}\tau$

for all $t\in[0,T]$, where
$v=v_{1}-v_{2}=\int_{0}^{T}u(\cdot,\tau)\mathrm{d}\tau.$ Noticing that,

$\displaystyle\|v\|^{2}_{L^{2}(\Omega)}=\int_{\Omega}\Big{|}\int_{0}^{T}u(x,\tau)\mathrm{d}\tau\Big{|}^{2}\mathrm{d}x\leq
T\int_{0}^{T}\|u(\cdot,\tau)\|^{2}_{L^{2}(\Omega)}\mathrm{d}\tau,$

we get

$\displaystyle\int_{0}^{t}\int_{\Omega}|v(x)|\,|u(x,\tau)|\,\mathrm{d}x\mathrm{d}\tau$
$\displaystyle\leq
T\Big{(}\int_{0}^{T}\|u(\cdot,\tau)\|^{2}_{L^{2}(\Omega)}\,\mathrm{d}\tau\Big{)}^{1/2}\,\Big{(}\int_{0}^{t}\|u(\cdot,\tau)\|^{2}_{L^{2}(\Omega)}\,\mathrm{d}\tau\Big{)}^{1/2},$

Therefore, we obtain the following inequality for all $t\in[0,T]$

$\displaystyle\displaystyle\|u(\cdot,t)\|^{2}_{L^{2}(\Omega)}\leq
2\kappa\Lambda
T\Big{(}\int_{0}^{T}\|u(\cdot,\tau)\|^{2}_{L^{2}(\Omega)}\,\mathrm{d}\tau\Big{)}^{1/2}\Big{(}\int_{0}^{t}\|u(\cdot,\tau)\|^{2}_{L^{2}(\Omega)}\,\mathrm{d}\tau\Big{)}^{1/2}.$
(4.4)

In short we rewrite the above inequality as follows

$\displaystyle\varrho^{\prime}(t)\leq 2\kappa\Lambda
T\varrho^{1/2}(T)\,\varrho^{1/2}(t)\quad\text{with}\quad\varrho(t)=\int_{0}^{t}\|u(\cdot,\tau)\|^{2}_{L^{2}(\Omega)}\,\mathrm{d}\tau.$

A routine integration yields that $\varrho^{1/2}(t)\leq\kappa\Lambda
T^{2}\varrho^{1/2}(T)$ and, in particular, $\varrho^{1/2}(T)\leq\kappa\Lambda
T^{2}\varrho^{1/2}(T)$. The latter inequality holds true only if
$\varrho(T)=0$ since $\kappa\Lambda T^{2}<1$, which implies that $u=0$. ∎

We now point out the following the closing remark which shows how the function
spaces considered in this note extends our studies to a sightly different type
of problems.

######  Remark 4.5.

Analogous results to those obtained in this notes can be established replacing
the Dirichlet complement condition $u=0\mbox{ in
}\;(\mathbb{R}^{N}\setminus\Omega)\times(0,T),$ the problem 1.1 with the
Neumann complement condition $\mathcal{N}u=0\mbox{ in
}\;(\mathbb{R}^{N}\setminus\Omega)\times(0,T),$ where $\mathcal{N}u$
represents the nonlocal normal derivative of $u$ across as defined in (2.8).
To this end, it is decisive to taking into account the setting of Theorem 2.2,
namely that $\Omega$ is bounded and Lipschitz and that $\nu$ satisfies the
asymptotic condition (2.3), in such a way that the compactness of the
embedding $V_{\nu}(\Omega|\mathbb{R}^{N})\hookrightarrow L^{2}(\Omega)$ holds
true. Wherefrom, one readily obtains (see [Fog20, FK22]) the Poincaré type
inequality

$\displaystyle\|u\|^{2}_{L^{2}(\Omega)}\leq C\mathcal{E}(u,u)\qquad\text{for
all $u\in V_{\nu}(\Omega|\mathbb{R}^{N})^{\perp}$},$

for some constant $C>0$ and where
$V_{\nu}(\Omega|\mathbb{R}^{N})^{\perp}=\big{\\{}V_{\nu}(\Omega|\mathbb{R}^{N})\mathrel{\mathop{\mathchar
58\relax}}\int_{\Omega}u\mathrm{d}x=0\big{\\}}$. These observations, alongside
of our procedure, allow to replace the space
$\mathbb{X}_{\nu}(\Omega|\mathbb{R}^{N})$ with the space
$V_{\nu}(\Omega|\mathbb{R}^{N})^{\perp}.$

Data Availability Statement (DAS): Data sharing not applicable, no datasets
were generated or analysed during the current study.

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