# Existence theory for a bushfire equation

Serena Dipierro  S.D., Department of Mathematics and Statistics, University of
Western Australia, 35 Stirling Highway, WA 6009 Crawley, Australia.
<EMAIL_ADDRESS>,  Enrico Valdinoci  E.V., Department of
Mathematics and Statistics, University of Western Australia, 35 Stirling
Highway, WA 6009 Crawley, Australia<EMAIL_ADDRESS>,  Glen
Wheeler  G. W., School of Mathematics and Applied Statistics, University of
Wollongong, Northfields Avenue, NSW 2500 Wollongong, Australia.
<EMAIL_ADDRESS>and  Valentina-Mira Wheeler  V.-M. W., School of Mathematics
and Applied Statistics, University of Wollongong, Northfields Avenue, NSW 2500
Wollongong, Australia<EMAIL_ADDRESS>
###### Abstract.

In [PAPER1] we have introduced a new model to describe front propagation in
bushfires. This model describes temperature diffusion in view of an ignition
process induced by an interaction kernel, the effect of the environmental wind
and that of the fire wind.

This has led to the introduction of a new partial differential equation of
evolutionary type, with nonlinear terms both of integral kind and involving
the gradient of the solution. This equation is, in a sense, “hybrid”, since it
encodes both analytical and geometric features of the front propagation. This
new characteristic makes the equation particularly interesting also from the
mathematical point of view, often falling outside the territory already
covered by standard methods.

In this paper, we start the mathematical treatment of this equation by
establishing the short time and global existence theory for this equation.

###### Key words and phrases:

Bushfire models, existence theory, fixed-point methods, interaction kernels,
pyrogenic flow.

###### 1991 Mathematics Subject Classification:

35A01, 47H10, 45H05, 35K57, 76R10.

Supported by Australian Research Council FL190100081 and DE190100379.

##  1\. Introduction

In this paper we consider a model for bushfire that we have recently
introduced in [PAPER1]. This model describes the spread of the fire on the
basis of relatively simple parameters, namely the environmental temperature,
the ignition temperature of the combustible, the environmental wind, and a
convective effect due to the temperature gradient in the vicinity of the flame
(known in the literature as “pyrogenic flow”).

The model has been confronted with real-world experiments, with numerical
simulations compared against data coming from real bushfires in Australia and
from lab experiments.

Since this is a new model, the mathematical theory underpinning it is still
under development: in this paper we aim at establishing an existence theory
for the partial differential equation at the core of this model.

This partial differential equation is of parabolic type, but it is not
completely standard. Therefore, on the one hand, it is tempting to utilize
functional analytic techniques, such as fixed-point arguments and iteration
methods, to demonstrate the existence of a solution, but, on the other hand,
these methods need to be adapted to the specific model under consideration,
which requires some bespoke argument and some ad hoc calculations.

Our bushfire model aims at describing the evolution of the environmental
temperature $u$ (which will correspond to the solution of the partial
differential equation under consideration). When the value of $u$ is above an
effective ignition temperature $\Theta$, the available fuel burns, producing a
forcing term that contributes to the equation through a site interaction
described by a kernel $K$ (roughly speaking, a burning site tends to increase
the temperature of its neighbors).

Two additional effects can be included in our model. First, the environmental
wind $\omega$ can contribute to the spreading of the fire. Also, an additional
effect due to the moving of the air can take part in this complex phenomenon,
on the basis of a convective term, which is modulated by a function $\beta$
(typically, $\beta$ is a function of the temperature, and one can think that
it is supported in the vicinity of the ignition temperature, since the
convective effects are expected to be more significant in a neighborhood of
the flame, where the gradient temperature is likely to be higher). These
additional effects accounting for the movement of the air cannot really stop
the propagation of a bushfire, instead they can contribute to its spread: for
this reason, these types of term are considered as active only when they point
“into the outward direction” of the front.

Though a full mathematical description and a precise list of hypotheses will
be given in the main results here below, in virtue of the above discussion,
the main equation that we consider takes the form

$\partial_{t}u=\Delta
u+\int_{\Omega}\big{(}u(y,t)-\Theta(y,t)\big{)}_{+}\,K(x,y)\,dy+\left(\left(\omega+\frac{\beta(u)\nabla
u}{|\nabla u|^{\gamma}}\right)\cdot\nabla u\right)_{-}\,,$

and our unknown function is $u=u(x,t)$.

For concreteness, we focus here on the case of homogeneous Dirichlet boundary
conditions (which is physically relevant, since, for example, Australia is an
island and bushfires are not expected to change, at least in a short time, the
temperature of the surrounding ocean). While we refer to [PAPER1] for a
comprehensive description of the model, let us mention that the partial
differential equation that we consider can also describe the moving front of
the fire, corresponding to the level sets of $u$ that match the ignition
temperature. This provides a “hybrid” model combining the solution of the
partial differential equation under consideration and a geometric term given
by the mean curvature of the moving front (and, in this setting, the
parabolicity of the partial differential equation entails that the equation of
the moving front presents the mean curvature term with “the right sign”
allowing for short time smoothing effects).

Of course, bushfire modeling has a long-standing tradition and several
different models and algorithms have been proposed to deal with different
aspects of fire ignition and propagation. Though an exhaustive list of
different models would go well beyond the scopes of this article, we refer to
the surveys in [BAK, PERRY, PASTOR] for the description of several models and
perspectives, as well as to [SMITH, HILTON201812] and the reference therein
for additional information on the pyrogenic flow.

We now dive into the technical description of our main results. The first
result that we present deals with short time existence and goes as follows:

######  Theorem 1.1 (Short time existence for a bushfire equation).

Let $\Omega$ be a bounded domain of $\mathds{R}^{n}$ with $C^{2}$ boundary.
Let also $K\in L^{2}(\Omega\times\Omega)$, $\Theta\in
L^{\infty}(\Omega\times(0,1))$, $\omega\in L^{\infty}(\Omega\times(0,1))$,
$\beta\in W^{1,\infty}(\mathds{R})$, and $\gamma\in(1,2]$.

Then, there exists $T_{\star}\in(0,1]$, depending only on $n$, $\Omega$,
$\|K\|_{L^{2}(\Omega\times\Omega)}$,
$\|\omega\|_{L^{\infty}(\Omega\times(0,1))}$, and
$\|\beta\|_{W^{1,\infty}(\mathds{R})}$, such that if $T\in(0,T_{\star}]$ then,
given $g\in H^{1}_{0}(\Omega)$, the problem

$\begin{cases}\partial_{t}u(x,t)=\Delta
u+\displaystyle\int_{\Omega}\big{(}u(y,t)-\Theta(y,t)\big{)}_{+}\,K(x,y)\,dy\\\
\qquad\qquad\qquad\qquad+\displaystyle\left(\left(\omega(x,t)+\frac{\beta(u(x,t))\nabla
u(x,t)}{|\nabla u(x,t)|^{\gamma}}\right)\cdot\nabla u(x,t)\right)_{-}\\\
\qquad{\mbox{ for $(x,t)\in\Omega\times(0,T)$,}}\\\ \\\ u(x,t)=0\\\
\qquad{\mbox{ for $(x,t)\in(\partial\Omega)\times(0,T)$,}}\\\ \\\
u(x,0)=g(x)\\\ \qquad{\mbox{ for $x\in\Omega$}}\end{cases}$

possesses111For simplicity, the notion of solution adopted in this paper is
the weak one, see e.g. [MR2597943, page 352]. a solution $u\in
L^{2}((0,T),\,H^{1}_{0}(\Omega))$.

The proof that we provide of Theorem 1.1 also allows for several
generalizations: for instance, one considers the case of a bounded function
$\beta$ also depending on $x$ and $t$, and Lipschitz continuous in its
dependence on $u$, as well as the case of a general second order parabolic
operator (e.g., in the setting of [MR2597943, Chapter 7]); here we do not go
into all the details of these generalizations, to keep things as simple as
possible.

We also point out that the technical condition $\gamma\in(1,2]$ corresponds to
a requirement on the relative importance of wind and convection in the
equation under consideration: roughly speaking, while the dependence on the
wind is bounded by a term scaling linearly with respect to the gradient of the
solution, the convective term is bounded by a term of the type $|\nabla
u|^{2-\gamma}$, hence sublinear when $\gamma\in(1,2]$ (stating, in some sense,
that the impact of the wind is assumed to be dominant with respect to that of
the convection).

The case $\gamma=1$ in which wind and convection scale the same is also of
interest. When $\beta$ is independent of $u$ (but possibly depending on space
and time, as mentioned above), the proof of Theorem 1.1 would go through also
when $\gamma=1$.

Moreover, under a smallness assumption on the nonlinear terms, the case of
$\beta$ depending on $u$ and $\gamma=1$ may be recovered through a limit
procedure, as addressed by the following result:

######  Theorem 1.2 (Short time existence for a bushfire equation with
convective terms with linear scaling).

Let $\Omega$ be a bounded domain of $\mathds{R}^{n}$ with $C^{2}$ boundary.
Let also $K\in L^{2}(\Omega\times\Omega)$, $\Theta\in
L^{\infty}(\Omega\times(0,1))$, $\omega\in L^{\infty}(\Omega\times(0,1))$ and
$\beta\in W^{1,\infty}(\mathds{R})$.

Then, there exist $\epsilon_{0}\in(0,1)$, depending only on $n$, $\Omega$ and
$\|\Theta_{-}\|_{L^{\infty}(\Omega\times(0,1))}$, and $T_{\star}\in(0,1]$,
depending only on $n$, $\Omega$, $\|K\|_{L^{2}(\Omega\times\Omega)}$,
$\|\omega\|_{L^{\infty}(\Omega\times(0,1))}$, and
$\|\beta\|_{W^{1,\infty}(\mathds{R})}$, such that if

$\|K\|_{L^{2}(\Omega\times\Omega)}+\|\omega\|_{L^{\infty}(\Omega\times(0,1))}+\|\beta\|_{L^{\infty}(\mathds{R})}\leqslant\epsilon_{0}$

and $T\in(0,T_{\star}]$ then, given $g\in H^{1}_{0}(\Omega)$, the problem

$\begin{cases}\partial_{t}u(x,t)=\Delta
u+\displaystyle\int_{\Omega}\big{(}u(y,t)-\Theta(y,t)\big{)}_{+}\,K(x,y)\,dy\\\
\qquad\qquad\qquad\qquad+\displaystyle\left(\left(\omega(x,t)+\frac{\beta(u(x,t))\nabla
u(x,t)}{|\nabla u(x,t)|}\right)\cdot\nabla u(x,t)\right)_{-}\\\ \qquad{\mbox{
for $(x,t)\in\Omega\times(0,T)$,}}\\\ \\\ u(x,t)=0\\\ \qquad{\mbox{ for
$(x,t)\in(\partial\Omega)\times(0,T)$,}}\\\ \\\ u(x,0)=g(x)\\\ \qquad{\mbox{
for $x\in\Omega$}}\end{cases}$

possesses a solution $u\in L^{2}((0,T),\,H^{1}_{0}(\Omega))$.

From Theorem 1.1, under some uniform assumption on the data, one can also
obtain a global in time solution, according to the following result:

######  Theorem 1.3 (Global in time existence for a bushfire equation).

Let $\Omega$ be a bounded domain of $\mathds{R}^{n}$ with $C^{2}$ boundary.
Let also $K\in L^{2}(\Omega\times\Omega)$, $\Theta\in
L^{\infty}(\Omega\times(0,+\infty))$, $\omega\in
L^{\infty}(\Omega\times(0,+\infty))$, $\beta\in W^{1,\infty}(\mathds{R})$, and
$\gamma\in(1,2]$.

Then, given $g\in H^{1}_{0}(\Omega)$, the problem

$\begin{cases}\partial_{t}u(x,t)=\Delta
u+\displaystyle\int_{\Omega}\big{(}u(y,t)-\Theta(y,t)\big{)}_{+}\,K(x,y)\,dy\\\
\qquad\qquad\qquad\qquad+\displaystyle\left(\left(\omega(x,t)+\frac{\beta(u(x,t))\nabla
u(x,t)}{|\nabla u(x,t)|^{\gamma}}\right)\cdot\nabla u(x,t)\right)_{-}\\\
\qquad{\mbox{ for $(x,t)\in\Omega\times(0,+\infty)$,}}\\\ \\\ u(x,t)=0\\\
\qquad{\mbox{ for $(x,t)\in(\partial\Omega)\times(0,+\infty)$,}}\\\ \\\
u(x,0)=g(x)\\\ \qquad{\mbox{ for $x\in\Omega$}}\end{cases}$

possesses a solution $u\in L^{2}((0,+\infty),\,H^{1}_{0}(\Omega))$.

For completeness, let us mention that a global in time result similar to
Theorem 1.3 holds true, with the same proof, when $\gamma=1$, under a smaller
condition on the structural parameter (this would follow by leveraging Theorem
1.2 in lieu of Theorem 1.1).

The rest of the paper is devoted to the proof of the main results stated
above. Namely, the short time existence result in Theorem 1.1 will be proved
in Section 2 and its variant to comprise the case of linearly scaling
convective terms will be considered in Section 3, while Section 4 will be
devoted to the proof of Theorem 1.3.

##  2\. Proof of Theorem 1.1

###  2.1. Toolbox

This section collects a number of technical results which will come in handy
in Section 2.2 to provide an efficient setup for the fixed-point argument.

######  Lemma 2.1.

Let $\Omega$ be a bounded, Lipschitz domain of $\mathds{R}^{n}$. Let also
$\beta\in W^{1,\infty}(\mathds{R})$ and $\gamma\in(1,2]$.

Then, there exists a constant $C>0$, depending only on $n$, $\Omega$, and
$\|\beta\|_{W^{1,\infty}(\mathds{R})}$, such that, for every $u$, $v\in
H^{1}_{0}(\Omega)$,

$\int_{\Omega}|\beta(u(x))-\beta(v(x))|^{2}|\nabla
u(x)|^{2(2-\gamma)}\,dx\leqslant C\,\|\nabla
u\|_{L^{2}(\Omega)}^{2(2-\gamma)}\|u-v\|_{L^{2}(\Omega)}^{2(\gamma-1)}.$ (2.1)

###### Proof.

We use the Hölder Inequality with exponents $\frac{1}{\gamma-1}$ and
$\frac{1}{2-\gamma}$ to see that

$\begin{split}&\int_{\Omega}|\beta(u(x))-\beta(v(x))|^{2}|\nabla
u(x)|^{2(2-\gamma)}\,dx\\\
&\qquad\leqslant\left(\int_{\Omega}|\beta(u(x))-\beta(v(x))|^{\frac{2}{\gamma-1}}\,dx\right)^{\gamma-1}\left(\int_{\Omega}|\nabla
u(x)|^{2}\,dx\right)^{2-\gamma}.\end{split}$ (2.2)

Furthermore,

$\displaystyle|\beta(u(x))-\beta(v(x))|^{\frac{2}{\gamma-1}}=|\beta(u(x))-\beta(v(x))|^{\frac{4-2\gamma}{\gamma-1}}|\beta(u(x))-\beta(v(x))|^{2}$
$\displaystyle\qquad\leqslant\Big{(}|\beta(u(x))|+|\beta(v(x))|\Big{)}^{\frac{4-2\gamma}{\gamma-1}}\|\nabla\beta\|_{L^{\infty}(\mathds{R})}^{2}|u(x)-v(x)|^{2}$
$\displaystyle\qquad\leqslant
2^{\frac{4-2\gamma}{\gamma-1}}\|\beta\|_{L^{\infty}(\mathds{R})}^{\frac{4-2\gamma}{\gamma-1}}\|\nabla\beta\|_{L^{\infty}(\mathds{R})}^{2}|u(x)-v(x)|^{2}.$

This and (2.2) give (2.1), as desired. ∎

######  Lemma 2.2.

Let $\Theta\in L^{\infty}(\Omega)$, $K\in L^{2}(\Omega\times\Omega)$, $u\in
L^{2}(\Omega)$, and

$f_{1,u}(x):=\int_{\Omega}\big{(}u(y)-\Theta(y)\big{)}_{+}\,K(x,y)\,dy.$ (2.3)

Then,

$\|f_{1,u}\|^{2}_{L^{2}(\Omega)}\leqslant
2\|K\|^{2}_{L^{2}(\Omega\times\Omega)}\,\left(\|u\|_{L^{2}(\Omega)}^{2}+|\Omega|\sup_{\Omega}\Theta_{-}^{2}\right),$

where $|\Omega|$ is the Lebesgue measure of $\Omega$.

###### Proof.

If $u(y)\geqslant\Theta(y)$ then

$0\leqslant\big{(}u(y)-\Theta(y)\big{)}_{+}=u(y)-\Theta(y)\leqslant
u(y)+\sup_{\Omega}\Theta_{-}.$

As a result, for all $y\in\Omega$,

$\big{(}u(y)-\Theta(y)\big{)}_{+}^{2}\leqslant\left(u(y)+\sup_{\Omega}\Theta_{-}\right)^{2}\leqslant
2\left(u^{2}(y)+\sup_{\Omega}\Theta_{-}^{2}\right)$

and, that being so, using the Cauchy-Schwarz Inequality we obtain that

$\displaystyle\left(\int_{\Omega}\big{(}u(y)-\Theta(y)\big{)}_{+}\,K(x,y)\,dy\right)^{2}\leqslant\int_{\Omega}\big{(}u(y)-\Theta(y)\big{)}_{+}^{2}\,dy\,\int_{\Omega}K^{2}(x,y)\,dy$
$\displaystyle\qquad\leqslant
2\left(\int_{\Omega}u^{2}(y)\,dy+|\Omega|\sup_{\Omega}\Theta_{-}^{2}\right)\,\int_{\Omega}K^{2}(x,y)\,dy.$

Integrating for $x\in\Omega$, the desired result follows. ∎

######  Lemma 2.3.

Let $\omega\in L^{\infty}(\Omega)$, $\beta\in L^{\infty}(\mathds{R})$,
$\gamma\in(1,2]$, and $u\in H^{1}(\Omega)$.

Let also

$f_{2,u}(x):=\left(\left(\omega(x)+\frac{\beta(u(x))\nabla u(x)}{|\nabla
u(x)|^{\gamma}}\right)\cdot\nabla u(x)\right)_{-}\,.$ (2.4)

Then,

$\|f_{2,u}\|_{L^{2}(\Omega)}\leqslant\sqrt{2}\left(\|\omega\|_{L^{\infty}(\Omega)}\,\|\nabla
u\|_{L^{2}(\Omega)}+\|\beta\|_{L^{\infty}(\mathds{R})}\,|\Omega|^{\frac{\gamma-1}{2}}\|\nabla
u\|_{L^{2}(\Omega)}^{2-\gamma}\right).$

###### Proof.

We have that

$\displaystyle|f_{2,u}(x)|$ $\displaystyle\leqslant$
$\displaystyle|\omega(x)|\,|\nabla u(x)|+|\beta(u(x))|\,|\nabla
u(x)|^{2-\gamma}$ $\displaystyle\leqslant$
$\displaystyle\|\omega\|_{L^{\infty}(\Omega)}\,|\nabla
u(x)|+\|\beta\|_{L^{\infty}(\mathds{R})}\,|\nabla u(x)|^{2-\gamma}.$

Therefore, squaring and integrating over $\Omega$,

$\|f_{2,u}\|_{L^{2}(\Omega)}^{2}\leqslant
2\left(\|\omega\|_{L^{\infty}(\Omega)}^{2}\,\|\nabla
u(x)\|_{L^{2}(\Omega)}^{2}+\|\beta\|_{L^{\infty}(\mathds{R})}^{2}\,\int_{\Omega}|\nabla
u(x)|^{2(2-\gamma)}\,dx\right).$ (2.5)

Now, if $\gamma=2$, from (2.5) we obtain that

$\|f_{2,u}\|_{L^{2}(\Omega)}\leqslant\sqrt{2}\left(\|\omega\|_{L^{\infty}(\Omega)}\,\|\nabla
u(x)\|_{L^{2}(\Omega)}+\|\beta\|_{L^{\infty}(\mathds{R})}\,\sqrt{|\Omega|}\right).$
(2.6)

If instead $\gamma\in(1,2)$, we exploit the Hölder Inequality with exponents
$\frac{1}{2-\gamma}$ and $\frac{1}{\gamma-1}$ to see that

$\displaystyle\int_{\Omega}|\nabla
u(x)|^{2(2-\gamma)}\,dx\leqslant|\Omega|^{\gamma-1}\left(\int_{\Omega}|\nabla
u(x)|^{2}\,dx\right)^{2-\gamma}=|\Omega|^{\gamma-1}\|\nabla
u\|_{L^{2}(\Omega)}^{2(2-\gamma)}.$

Plugging this information into (2.5) we conclude that, in this case,

$\|f_{2,u}\|_{L^{2}(\Omega)}\leqslant\sqrt{2}\left(\|\omega\|_{L^{\infty}(\Omega)}\,\|\nabla
u(x)\|_{L^{2}(\Omega)}+\|\beta\|_{L^{\infty}(\mathds{R})}\,|\Omega|^{\frac{\gamma-1}{2}}\|\nabla
u\|_{L^{2}(\Omega)}^{2-\gamma}\right).$

From this and (2.6) the desired result follows. ∎

######  Lemma 2.4.

Let $\Theta\in L^{\infty}(\Omega)$, $K\in L^{2}(\Omega\times\Omega)$, and $u$,
$v\in L^{2}(\Omega)$.

Then, in the notation of (2.3),

$\|f_{1,u}-f_{1,v}\|_{L^{2}(\Omega)}\leqslant\|K\|_{L^{2}(\Omega\times\Omega)}\|u-v\|_{L^{2}(\Omega)}.$

###### Proof.

Since the positive part is a Lipschitz function, with Lipschitz constant
controlled by $1$, we have that, for all $a$, $b\in\mathds{R}$,

$|a_{+}-b_{+}|\leqslant|a-b|.$

In particular, taking $a:=u(y)-\Theta(y)$ and $b:=v(y)-\Theta(y)$, we see that

$\Big{|}\big{(}u(y)-\Theta(y)\big{)}_{+}-\big{(}v(y)-\Theta(y)\big{)}_{+}\Big{|}\leqslant|u(y)-v(y)|.$

On this account,

$\displaystyle|f_{1,u}(x)-f_{1,v}(x)|$ $\displaystyle=$
$\displaystyle\left|\int_{\Omega}\Big{(}\big{(}u(y)-\Theta(y)\big{)}_{+}-\big{(}v(y)-\Theta(y)\big{)}_{+}\Big{)}\,K(x,y)\,dy\right|$
$\displaystyle\leqslant$
$\displaystyle\int_{\Omega}\Big{|}\big{(}u(y)-\Theta(y)\big{)}_{+}-\big{(}v(y)-\Theta(y)\big{)}_{+}\Big{|}\,|K(x,y)|\,dy$
$\displaystyle\leqslant$
$\displaystyle\int_{\Omega}|u(y)-v(y)|\,|K(x,y)|\,dy.$

Using the Hölder Inequality, we conclude that

$|f_{1,u}(x)-f_{1,v}(x)|^{2}\leqslant\int_{\Omega}|u(y)-v(y)|^{2}\,dy\int_{\Omega}K^{2}(x,y)\,dy$

and the desired result follows by integrating for $x\in\Omega$. ∎

######  Lemma 2.5.

Let $\Omega$ be a bounded, Lipschitz domain of $\mathds{R}^{n}$. Let
$\omega\in L^{\infty}(\Omega)$, $\beta\in W^{1,\infty}(\mathds{R})$, and
$\gamma\in(1,2]$.

Then, there exist a constant $C>0$, depending only on $n$, $\Omega$,
$\|\omega\|_{L^{\infty}(\Omega)}$, and $\|\beta\|_{W^{1,\infty}(\mathds{R})}$,
such that, in the notation of (2.4), for every $u$, $v\in H^{1}_{0}(\Omega)$
it holds that

$\|f_{2,u}-f_{2,v}\|_{L^{2}(\Omega)}\leqslant\begin{cases}C\Big{(}\|\nabla
u-\nabla v\|_{L^{2}(\Omega)}+\|\nabla u-\nabla
v\|_{L^{2}(\Omega)}^{2-\gamma}+\|\nabla
u\|_{L^{2}(\Omega)}^{2-\gamma}\,\|u-v\|_{L^{2}(\Omega)}^{\gamma-1}\Big{)}\\\
\qquad\qquad\qquad{\mbox{ if~{}$\gamma\in(1,2)$,}}\\\
C\|u-v\|_{H^{1}(\Omega)}\quad{\mbox{ if~{}$\gamma=2$.}}\end{cases}$

###### Proof.

We observe that the negative part is a Lipschitz function, with Lipschitz
constant controlled by $1$, whence, for all $a$, $b\in\mathds{R}$,

$|a_{-}-b_{-}|\leqslant|a-b|.$

In particular, taking

$\displaystyle a:=\left(\omega(x)+\frac{\beta(u(x))\nabla u(x)}{|\nabla
u(x)|^{\gamma}}\right)\cdot\nabla u(x)$ and $\displaystyle
b:=\left(\omega(x)+\frac{\beta(v(x))\nabla v(x)}{|\nabla
v(x)|^{\gamma}}\right)\cdot\nabla v(x),$

we find that

$\begin{split}&|f_{2,u}(x)-f_{2,v}(x)|\\\
&\quad=\left|\left(\left(\omega(x)+\frac{\beta(u(x))\nabla u(x)}{|\nabla
u(x)|^{\gamma}}\right)\cdot\nabla
u(x)\right)_{-}-\left(\left(\omega(x)+\frac{\beta(v(x))\nabla v(x)}{|\nabla
v(x)|^{\gamma}}\right)\cdot\nabla v(x)\right)_{-}\right|\\\
&\quad\leqslant\left|\left(\omega(x)+\frac{\beta(u(x))\nabla u(x)}{|\nabla
u(x)|^{\gamma}}\right)\cdot\nabla u(x)-\left(\omega(x)+\frac{\beta(v(x))\nabla
v(x)}{|\nabla v(x)|^{\gamma}}\right)\cdot\nabla v(x)\right|\\\
&\quad\leqslant|\omega(x)|\,|\nabla u(x)-\nabla
v(x)|+\Big{|}\beta(u(x))|\nabla u(x)|^{2-\gamma}-\beta(v(x))|\nabla
v(x)|^{2-\gamma}\Big{|}\\\
&\quad\leqslant\|\omega\|_{L^{\infty}(\Omega)}|\nabla u(x)-\nabla v(x)|\\\
&\qquad+|\beta(v(x))|\,\Big{|}|\nabla u(x)|^{2-\gamma}-|\nabla
v(x)|^{2-\gamma}\Big{|}+\Big{|}\big{(}\beta(u(x))-\beta(v(x))\big{)}|\nabla
u(x)|^{2-\gamma}\Big{|}\\\
&\quad\leqslant\|\omega\|_{L^{\infty}(\Omega)}|\nabla u(x)-\nabla v(x)|\\\
&\qquad+\|\beta\|_{L^{\infty}(\mathds{R})}\,\Big{|}|\nabla
u(x)|^{2-\gamma}-|\nabla
v(x)|^{2-\gamma}\Big{|}+\big{|}\beta(u(x))-\beta(v(x))\big{|}\,|\nabla
u(x)|^{2-\gamma}.\end{split}$

Notice that if $\gamma=2$ then $\Big{|}|\nabla u(x)|^{2-\gamma}-|\nabla
v(x)|^{2-\gamma}\Big{|}=0$. Hence, from now on we suppose that the second term
in the last inequality above is present only when $\gamma\in(1,2)$, otherwise
it is set to be zero in the following computations.

Accordingly,

$\begin{split}&\int_{\Omega}|f_{2,u}(x)-f_{2,v}(x)|^{2}\,dx\\\
\leqslant\;&3\|\omega\|_{L^{\infty}(\Omega)}^{2}\int_{\Omega}|\nabla
u(x)-\nabla
v(x)|^{2}\,dx+3\|\beta\|_{L^{\infty}(\mathds{R})}^{2}\int_{\Omega}\Big{|}|\nabla
u(x)|^{2-\gamma}-|\nabla v(x)|^{2-\gamma}\Big{|}^{2}\,dx\\\
&\qquad+3\int_{\Omega}\big{|}\beta(u(x))-\beta(v(x))\big{|}^{2}\,|\nabla
u(x)|^{2(2-\gamma)}\,dx.\end{split}$ (2.7)

We observe that, for all $a$, $b>0$,

$\big{|}a^{2-\gamma}-b^{2-\gamma}\big{|}\leqslant|a-b|^{2-\gamma}.$ (2.8)

To check this, we consider the function $t\in(0,1)\mapsto
g(t):=\frac{1-t^{2-\gamma}}{(1-t)^{2-\gamma}}$ and we point out that $g(0)=1$
and

$\displaystyle\lim_{t\to 1}g(t)=\lim_{t\to
1}\frac{1-t^{2-\gamma}}{(1-t)^{2-\gamma}}=\lim_{t\to
1}\frac{-(2-\gamma)t^{1-\gamma}}{(2-\gamma)(1-t)^{1-\gamma}}=0.$

Moreover,

$\displaystyle g^{\prime}(t)$ $\displaystyle=$
$\displaystyle\frac{-(2-\gamma)t^{1-\gamma}(1-t)^{2-\gamma}+(2-\gamma)(1-t^{2-\gamma})(1-t)^{1-\gamma}}{(1-t)^{2(2-\gamma)}}$
$\displaystyle=$
$\displaystyle\frac{(2-\gamma)(1-t^{1-\gamma})}{(1-t)^{3-\gamma}}$
$\displaystyle\leqslant$ $\displaystyle 0.$

Gathering these pieces of information, we obtain that $g(t)\leqslant 1$ for
all $t\in(0,1)$. Namely, for all $t\in(0,1)$,

$1-t^{2-\gamma}\leqslant(1-t)^{2-\gamma}.$ (2.9)

Now, we suppose, without loss of generality, that $a>b$ and we write

$\big{|}a^{2-\gamma}-b^{2-\gamma}\big{|}=a^{2-\gamma}-b^{2-\gamma}=a^{2-\gamma}\left(1-t^{2-\gamma}\right),$

with $t:=b/a\in(0,1)$. Therefore, we deduce from (2.9) that

$\big{|}a^{2-\gamma}-b^{2-\gamma}\big{|}\leqslant
a^{2-\gamma}(1-t)^{2-\gamma}=(a-b)^{2-\gamma},$

which establishes (2.8).

Using (2.8) with $a:=|\nabla u(x)|$ and $b:=|\nabla v(x)|$, we conclude that

$\Big{|}|\nabla u(x)|^{2-\gamma}-|\nabla
v(x)|^{2-\gamma}\Big{|}^{2}\leqslant\Big{|}|\nabla u(x)|-|\nabla
v(x)|\Big{|}^{2(2-\gamma)}\leqslant\Big{|}\nabla u(x)-\nabla
v(x)\Big{|}^{2(2-\gamma)}$ (2.10)

In light of this and Lemma 2.1, we deduce from (2.7) that

$\begin{split}&\|f_{2,u}-f_{2,v}\|^{2}_{L^{2}(\Omega)}\\\
\leqslant\;&C\Bigg{[}\|\omega\|_{L^{\infty}(\Omega)}^{2}\|\nabla u-\nabla
v\|_{L^{2}(\Omega)}^{2}+\|\beta\|_{L^{\infty}(\mathds{R})}^{2}\int_{\Omega}\Big{|}\nabla
u(x)-\nabla v(x)\Big{|}^{2(2-\gamma)}\,dx\\\ &\qquad+\|\nabla
u\|_{L^{2}(\Omega)}^{2(2-\gamma)}\|u-v\|_{L^{2}(\Omega)}^{2(\gamma-1)}\Bigg{]}.\end{split}$
(2.11)

Now we exploit the Hölder Inequality with exponents $\frac{1}{2-\gamma}$ and
$\frac{1}{\gamma-1}$ and see that

$\int_{\Omega}\Big{|}\nabla u(x)-\nabla
v(x)\Big{|}^{2(2-\gamma)}\,dx\leqslant|\Omega|^{\gamma-1}\left(\int_{\Omega}\Big{|}\nabla
u(x)-\nabla v(x)\Big{|}^{2}\,dx\right)^{2-\gamma}.$

Plugging this information into (2.11) entails the desired result, up to
renaming constants. ∎

###  2.2. The fixed-point argument

Here we consider a bounded domain $\Omega\subset\mathds{R}^{n}$ with Lipschitz
boundary, we let $T>0$, and we consider the functional spaces

$X:=L^{2}((0,T),\,H^{1}_{0}(\Omega))\qquad{\mbox{and}}\qquad
Y:=L^{2}((0,T),\,L^{2}(\Omega)).$

See e.g. [MR916688] and the references therein for classical results about the
theory of Sobolev spaces of Hilbert and Banach space-valued functions.

Also, we take $g\in H^{1}_{0}(\Omega)$.

For every $f\in Y$, we define $\sigma_{f}=\sigma_{f}(x,t)$ to be the solution
of the parabolic problem

$\begin{cases}\partial_{t}\sigma_{f}(x,t)=\Delta\sigma_{f}(x,t)+f(x,t)&{\mbox{
for $(x,t)\in\Omega\times(0,T)$,}}\\\ \sigma_{f}(x,t)=0&{\mbox{ for
$(x,t)\in(\partial\Omega)\times(0,T)$,}}\\\ \sigma_{f}(x,0)=g(x)&{\mbox{ for
$x\in\Omega$.}}\end{cases}$

We recall that the above solution indeed exists (say, as a weak solution, see
Theorem 3 on page 356 of [MR2597943]) and it is unique (see Theorem 4 on page
358 of [MR2597943]).

Additionally (see Theorem 5 on page 360 of [MR2597943]) we have the regularity
estimate

$\begin{split}&\sup_{t\in[0,T]}\|\sigma_{f}(\cdot,t)\|_{H^{1}(\Omega)}+\|\sigma_{f}\|_{L^{2}((0,T),\,H^{2}(\Omega))}+\|\partial_{t}\sigma_{f}\|_{L^{2}((0,T),\,L^{2}(\Omega))}\\\
&\qquad\leqslant
C\Big{(}\|f\|_{L^{2}((0,T),\,L^{2}(\Omega))}+\|g\|_{H^{1}(\Omega)}\Big{)},\end{split}$
(2.12)

for some $C>0$ depending on $n$, $\Omega$, and $T$.

We remark that:

######  Lemma 2.6.

When $T\in(0,1]$ the constant $C$ in (2.12) depends on $n$ and $\Omega$, but
not on $T$.

###### Proof.

We define $\widetilde{f}(x,t):=f(x,t)\chi_{[0,T]}(t)$ and notice that, by the
uniqueness of the solution and the fact that $\widetilde{f}=f$ when
$t\in[0,T]$, we have $\sigma_{\widetilde{f}}=\sigma_{f}$ when $t\in[0,T)$. We
also observe that, for all $T\in(0,1]$,

$\|\widetilde{f}\|_{L^{2}((0,1),\,L^{2}(\Omega))}^{2}=\iint_{\Omega\times(0,1)}|\widetilde{f}(x,t)|^{2}\,dx\,dt=\iint_{\Omega\times(0,T)}|f(x,t)|^{2}\,dx\,dt=\|f\|_{L^{2}((0,T),\,L^{2}(\Omega))}^{2}.$

As a consequence, we can apply (2.12) for $T:=1$ and obtain that

$\begin{split}&\sup_{t\in[0,1]}\|\sigma_{\widetilde{f}}(\cdot,t)\|_{H^{1}(\Omega)}+\|\sigma_{\widetilde{f}}\|_{L^{2}((0,1),\,H^{2}(\Omega))}+\|\partial_{t}\sigma_{\widetilde{f}}\|_{L^{2}((0,1),\,L^{2}(\Omega))}\\\
&\qquad\leqslant
C\Big{(}\|\widetilde{f}\|_{L^{2}((0,1),\,L^{2}(\Omega))}+\|g\|_{H^{1}(\Omega)}\Big{)}\\\
&\qquad=C\Big{(}\|f\|_{L^{2}((0,T),\,L^{2}(\Omega))}+\|g\|_{H^{1}(\Omega)}\Big{)},\end{split}$

with $C>0$ depending only on $n$ and $\Omega$.

This and the fact that, for all $T\in(0,1]$,

$\displaystyle\sup_{t\in[0,T]}\|\sigma_{f}(\cdot,t)\|_{H^{1}(\Omega)}+\|\sigma_{f}\|_{L^{2}((0,T),\,H^{2}(\Omega))}+\|\partial_{t}\sigma_{f}\|_{L^{2}((0,T),\,L^{2}(\Omega))}$
$\displaystyle\qquad=\sup_{t\in[0,T]}\|\sigma_{\widetilde{f}}(\cdot,t)\|_{H^{1}(\Omega)}+\|\sigma_{\widetilde{f}}\|_{L^{2}((0,T),\,H^{2}(\Omega))}+\|\partial_{t}\sigma_{\widetilde{f}}\|_{L^{2}((0,T),\,L^{2}(\Omega))}$
$\displaystyle\qquad\leqslant\sup_{t\in[0,1]}\|\sigma_{\widetilde{f}}(\cdot,t)\|_{H^{1}(\Omega)}+\|\sigma_{\widetilde{f}}\|_{L^{2}((0,1),\,H^{2}(\Omega))}+\|\partial_{t}\sigma_{\widetilde{f}}\|_{L^{2}((0,1),\,L^{2}(\Omega))}$

give the desired result. ∎

Now we consider functions $K\in L^{2}(\Omega\times\Omega)$, $\Theta\in
L^{\infty}(\Omega\times(0,T))$, $\omega\in L^{\infty}(\Omega\times(0,T))$, and
$\beta\in W^{1,\infty}(\mathds{R})$. We also let $\gamma\in(1,2]$.

Given $u\in X$, we define

$\begin{split}f_{u}(x,t)&:=\int_{\mathds{R}^{n}}\big{(}u(y,t)-\Theta(y,t)\big{)}_{+}\,K(x,y)\,dy\\\
&\qquad\qquad+\left(\left(\omega(x,t)+\frac{\beta(u(x,t))\nabla
u(x,t)}{|\nabla u(x,t)|^{\gamma}}\right)\cdot\nabla
u(x,t)\right)_{-}.\end{split}$ (2.13)

In this setting, we have:

######  Lemma 2.7.

It holds that $f_{u}\in Y$, with

$\|f_{u}\|_{Y}\leqslant
C\,\left(\|u\|_{X}+\|u\|_{X}^{2-\gamma}+T\sup_{\Omega\times(0,T)}\Theta_{-}\right),$

for a suitable $C>0$ depending only on $n$, $\Omega$,
$\|K\|_{L^{2}(\Omega\times\Omega)}$,
$\|\omega\|_{L^{\infty}(\Omega\times(0,T))}$, and
$\|\beta\|_{W^{1,\infty}(\mathds{R})}$.

###### Proof.

For a given $t\in(0,T)$, after (2.3) and (2.4), we have that

$f_{u}(x,t)=f_{1,u^{(t)}}(x)+f_{2,u^{(t)}}(x),\qquad{\mbox{where}}\qquad{u^{(t)}}(x):=u(x,t).$
(2.14)

By Lemmata 2.2 and 2.3, for any $t\in(0,T)$,

$\displaystyle\|f_{u}(\cdot,t)\|_{L^{2}(\Omega)}$ $\displaystyle\leqslant$
$\displaystyle\|f_{1,u^{(t)}}\|_{L^{2}(\Omega)}+\|f_{2,u^{(t)}}\|_{L^{2}(\Omega)}$
$\displaystyle\leqslant$ $\displaystyle
C\left(\|u^{(t)}\|_{H^{1}(\Omega)}+\|u^{(t)}\|_{H^{1}(\Omega)}^{2-\gamma}+\sup_{x\in\Omega}\Theta_{-}(x,t)\right),$

whence the desired result follows by squaring and integrating for $t\in(0,T)$.
∎

Now we consider the nonlinear operator

$X\ni u\longmapsto Au:=\sigma_{f_{u}}.$

We have that:

######  Lemma 2.8.

For all $u\in X$, we have that $Au\in X$, with

$\|Au\|_{X}\leqslant
C\left(\|u\|_{X}+\|u\|_{X}^{2-\gamma}+\|g\|_{H^{1}(\Omega)}+T\sup_{\Omega\times(0,T)}\Theta_{-}\right).$
(2.15)

Moreover, for all $u$, $v\in X$ with $\|u-v\|_{X}\leqslant 1$,

$\|Au-Av\|_{X}\leqslant C_{u}\,\|u-v\|_{X}^{\alpha}.$ (2.16)

Here above, the constants $C>0$ and $\alpha$ depend only on $n$, $\Omega$,
$T$, $\|K\|_{L^{2}(\Omega\times\Omega)}$,
$\|\omega\|_{L^{\infty}(\Omega\times(0,T))}$, and
$\|\beta\|_{W^{1,\infty}(\mathds{R})}$, while $C_{u}>0$ depends also on
$\|u\|_{X}$.

###### Proof.

Owing to (2.12),

$\displaystyle\|Au\|_{X}=\|\sigma_{f_{u}}\|_{L^{2}((0,T),\,H^{1}(\Omega))}\leqslant
C\Big{(}\|f_{u}\|_{L^{2}((0,T),\,L^{2}(\Omega))}+\|g\|_{H^{1}(\Omega)}\Big{)}$
$\displaystyle\qquad\qquad\qquad=C\Big{(}\|f_{u}\|_{Y}+\|g\|_{H^{1}(\Omega)}\Big{)}.$

This and Lemma 2.7 yield (2.15), as desired.

Now we let $\sigma_{\star}:=\sigma_{f_{u}}-\sigma_{f_{v}}$ and
$f_{\star}(x,t):=f_{u}(x,t)-f_{v}(x,t)$, and we see that

$\begin{cases}\partial_{t}\sigma_{\star}(x,t)=\Delta\sigma_{\star}(x,t)+f_{\star}(x,t)&{\mbox{
for $(x,t)\in\Omega\times(0,T)$,}}\\\ \sigma_{\star}(x,t)=0&{\mbox{ for
$(x,t)\in(\partial\Omega)\times(0,T)$,}}\\\ \sigma_{\star}(x,0)=0&{\mbox{ for
$x\in\Omega$.}}\end{cases}$

For this reason, by (2.12),

$\displaystyle\|\sigma_{\star}\|_{L^{2}((0,T),\,H^{2}(\Omega))}\leqslant
C\|f_{\star}\|_{L^{2}((0,T),\,L^{2}(\Omega))}=C\Big{(}\|f_{u}-f_{v}\|_{L^{2}((0,T),\,L^{2}(\Omega))}\Big{)}$

and therefore

$\begin{split}&\|Au-
Av\|_{X}=\|\sigma_{f_{u}}-\sigma_{f_{v}}\|_{L^{2}((0,T),\,H^{1}(\Omega))}=\|\sigma_{\star}\|_{L^{2}((0,T),\,H^{1}(\Omega))}\\\
&\qquad\leqslant
C\Big{(}\|f_{u}-f_{v}\|_{L^{2}((0,T),\,L^{2}(\Omega))}\Big{)}.\end{split}$
(2.17)

Now, in the notation of (2.3), (2.4) and (2.14), and recalling Lemmata 2.4 and
2.5, we see that, for every $t\in(0,T)$,

$\begin{split}&\|f_{u}(\cdot,t)-f_{v}(\cdot,t)\|_{L^{2}(\Omega)}\\\
&\quad=\|f_{1,u^{(t)}}+f_{2,u^{(t)}}-f_{1,v^{(t)}}-f_{2,v^{(t)}}\|_{L^{2}(\Omega)}\\\
&\quad\leqslant\|f_{1,u^{(t)}}-f_{1,v^{(t)}}\|_{L^{2}(\Omega)}+\|f_{2,u^{(t)}}-f_{2,v^{(t)}}\|_{L^{2}(\Omega)}\\\
&\quad\leqslant C\Big{(}\|\nabla
u^{(t)}\|_{L^{2}(\Omega)}^{2-\gamma}+1\Big{)}\Big{(}\|u^{(t)}-v^{(t)}\|_{H^{1}(\Omega)}+\|u^{(t)}-v^{(t)}\|_{L^{2}(\Omega)}^{\gamma-1}+\|\nabla
u^{(t)}-\nabla v^{(t)}\|_{L^{2}(\Omega)}^{2-\gamma}\Big{)}.\end{split}$

We point out that the last term in the above inequality is set to equal zero
when $\gamma=2$, as specified in Lemma 2.5.

Hence, for all $\epsilon>0$,

$\begin{split}&\|f_{u}(\cdot,t)-f_{v}(\cdot,t)\|_{L^{2}(\Omega)}^{2}\\\
&\qquad\leqslant C\epsilon\Big{(}\|\nabla
u^{(t)}\|_{L^{2}(\Omega)}^{2(2-\gamma)}+1\Big{)}\\\
&\qquad\qquad+\frac{C}{\epsilon}\Big{(}\|u^{(t)}-v^{(t)}\|_{H^{1}(\Omega)}^{2}+\|u^{(t)}-v^{(t)}\|_{L^{2}(\Omega)}^{2(\gamma-1)}+\|\nabla
u^{(t)}-\nabla v^{(t)}\|_{L^{2}(\Omega)}^{2(2-\gamma)}\Big{)}\\\
&\qquad\leqslant C\epsilon\Big{(}\|\nabla
u^{(t)}\|_{L^{2}(\Omega)}^{2}+1\Big{)}\\\
&\qquad\qquad+\frac{C}{\epsilon}\Big{(}\|u^{(t)}-v^{(t)}\|_{H^{1}(\Omega)}^{2}+\|u^{(t)}-v^{(t)}\|_{L^{2}(\Omega)}^{2(\gamma-1)}+\|\nabla
u^{(t)}-\nabla v^{(t)}\|_{L^{2}(\Omega)}^{2(2-\gamma)}\Big{)}\end{split}$

and thus, integrating in $t\in(0,T)$, and exploiting the Hölder Inequality,

$\begin{split}&\|f_{u}-f_{v}\|_{L^{2}((0,T),\,L^{2}(\Omega))}^{2}\\\
&\qquad\leqslant
C\epsilon\Big{(}\|u\|_{L^{2}((0,T),\,H^{1}(\Omega))}^{2}+1\Big{)}\\\
&\qquad\qquad+\frac{C}{\epsilon}\Big{(}\|u-v\|_{L^{2}((0,T),\,L^{2}(\Omega))}^{2}+T^{2-\gamma}\|u-v\|_{L^{2}((0,T),\,L^{2}(\Omega))}^{2(\gamma-1)}\\\
&\qquad\qquad\qquad\qquad+T^{\gamma-1}\|\nabla u-\nabla
v\|_{L^{2}((0,T),\,L^{2}(\Omega))}^{2(2-\gamma)}\Big{)}\\\ &\qquad\leqslant
C\epsilon\Big{(}\|u\|_{X}^{2}+1\Big{)}+\frac{C}{\epsilon}\Big{(}\|u-v\|_{X}^{2}+\|u-v\|_{X}^{2(\gamma-1)}+\|u-v\|_{X}^{2(2-\gamma)}\Big{)}\\\
&\qquad\leqslant
C\epsilon\Big{(}\|u\|_{X}^{2}+1\Big{)}+\frac{C}{\epsilon}\Big{(}\|u-v\|_{X}^{2(\gamma-1)}+\|u-v\|_{X}^{2(2-\gamma)}\Big{)}.\end{split}$

In particular, choosing

$\epsilon:=\sqrt{\frac{\|u-v\|_{X}^{2(\gamma-1)}+\|u-v\|_{X}^{2(2-\gamma)}}{\|u\|_{X}^{2}+1}},$

we gather that

$\displaystyle\|f_{u}-f_{v}\|_{L^{2}((0,T),\,L^{2}(\Omega))}^{2}$
$\displaystyle\leqslant$ $\displaystyle
C\sqrt{\|u\|_{X}^{2}+1}\;\sqrt{\|u-v\|_{X}^{2(\gamma-1)}+\|u-v\|_{X}^{2(2-\gamma)}}$
$\displaystyle\leqslant$ $\displaystyle
C_{u}\,\sqrt{\|u-v\|_{X}^{2(\gamma-1)}+\|u-v\|_{X}^{2(2-\gamma)}}$
$\displaystyle\leqslant$ $\displaystyle
C_{u}\,\Big{(}\|u-v\|_{X}^{\gamma-1}+\|u-v\|_{X}^{2-\gamma}\Big{)}.$

Up to renaming constants, this and (2.17) give (2.16), as desired. ∎

Now we put forth a useful compactness result.

######  Lemma 2.9.

Let $v_{k}\in X$, with

$\sup_{k\in\mathds{N}}\Big{(}\|v_{k}\|_{L^{2}((0,T),\,H^{2}(\Omega))}+\|\partial_{t}v_{k}\|_{L^{2}((0,T),\,L^{2}(\Omega))}\Big{)}<+\infty.$
(2.18)

Then, there exists $v\in X$ such that, up to a subsequence, $v_{k}\to v$ in
$X$ as $k\to+\infty$.

###### Proof.

By (2.18), up to a subsequence, we can assume that

$v_{k}$ converges weakly to some $\widetilde{v}\in
L^{2}((0,T),\,H^{1}_{0}(\Omega)\cap H^{2}(\Omega))$ as $k\to+\infty$ (2.19)

and, up to extracting a further subsequence, we can assume that

$v_{k}$ converges to some $v$ in $L^{2}((0,T),\,L^{2}(\Omega))$ as
$k\to+\infty$. (2.20)

We claim that

$v=\tilde{v}.$ (2.21)

To this end, in view of (2.19) we have that, for every $\phi\in
L^{2}((0,T),\,H^{1}_{0}(\Omega)\cap H^{4}(\Omega))$ with $\Delta\phi\in
L^{2}((0,T),\,H^{1}_{0}(\Omega))$,

$\begin{split}&\int_{0}^{T}\Big{(}\langle\widetilde{v}(\cdot,t),\phi(\cdot,t)\rangle_{L^{2}(\Omega)}-\langle\widetilde{v}(\cdot,t),\Delta\phi(\cdot,t)\rangle_{L^{2}(\Omega)}+\langle\widetilde{v}(\cdot,t),\Delta^{2}\phi(\cdot,t)\rangle_{L^{2}(\Omega)}\Big{)}\,dt\\\
=&\int_{0}^{T}\left(\langle\widetilde{v}(\cdot,t),\phi(\cdot,t)\rangle_{L^{2}(\Omega)}+\langle\nabla\widetilde{v}(\cdot,t),\nabla\phi(\cdot,t)\rangle_{L^{2}(\Omega)}+\sum_{i,j=1}^{n}\langle\partial_{x_{i}x_{j}}^{2}\widetilde{v}(\cdot,t),\partial_{x_{i}x_{j}}^{2}\phi(\cdot,t)\rangle_{L^{2}(\Omega)}\right)\,dt\\\
=&\lim_{k\to+\infty}\int_{0}^{T}\left(\langle
v_{k}(\cdot,t),\phi(\cdot,t)\rangle_{L^{2}(\Omega)}+\langle\nabla
v_{k}(\cdot,t),\nabla\phi(\cdot,t)\rangle_{L^{2}(\Omega)}+\sum_{i,j=1}^{n}\langle\partial_{x_{i}x_{j}}^{2}v_{k}(\cdot,t),\partial_{x_{i}x_{j}}^{2}\phi(\cdot,t)\rangle_{L^{2}(\Omega)}\right)\,dt\\\
=&\lim_{k\to+\infty}\int_{0}^{T}\Big{(}\langle
v_{k}(\cdot,t),\phi(\cdot,t)\rangle_{L^{2}(\Omega)}-\langle
v_{k}(\cdot,t),\Delta\phi(\cdot,t)\rangle_{L^{2}(\Omega)}+\langle
v_{k}(\cdot,t),\Delta^{2}\phi(\cdot,t)\rangle_{L^{2}(\Omega)}\Big{)}\,dt.\end{split}$

Hence, by (2.20),

$\displaystyle\int_{0}^{T}\Big{(}\langle\widetilde{v}(\cdot,t),\phi(\cdot,t)\rangle_{L^{2}(\Omega)}-\langle\widetilde{v}(\cdot,t),\Delta\phi(\cdot,t)\rangle_{L^{2}(\Omega)}+\langle\widetilde{v}(\cdot,t),\Delta^{2}\phi(\cdot,t)\rangle_{L^{2}(\Omega)}\Big{)}\,dt$
$\displaystyle=$ $\displaystyle\int_{0}^{T}\Big{(}\langle
v(\cdot,t),\phi(\cdot,t)\rangle_{L^{2}(\Omega)}-\langle
v(\cdot,t),\Delta\phi(\cdot,t)\rangle_{L^{2}(\Omega)}+\langle
v(\cdot,t),\Delta^{2}\phi(\cdot,t)\rangle_{L^{2}(\Omega)}\Big{)}\,dt.$

That is, letting $w:=v-\widetilde{v}$,

$\begin{split}&\int_{0}^{T}\left\langle
w(\cdot,t),\Big{(}\phi(\cdot,t)-\Delta\phi(\cdot,t)+\Delta^{2}\phi(\cdot,t)\Big{)}\right\rangle_{L^{2}(\Omega)}\,dt\\\
=\;&\int_{0}^{T}\Big{(}\langle
w(\cdot,t),\phi(\cdot,t)\rangle_{L^{2}(\Omega)}-\langle
w(\cdot,t),\Delta\phi(\cdot,t)\rangle_{L^{2}(\Omega)}+\langle
w(\cdot,t),\Delta^{2}\phi(\cdot,t)\rangle_{L^{2}(\Omega)}\Big{)}\,dt\\\
=\;&0.\end{split}$ (2.22)

Now we choose $\phi\in L^{2}((0,T),\,H^{1}_{0}(\Omega)\cap H^{2}(\Omega))$ as
the minimizer of the functional

$\iint_{\Omega\times(0,T)}\Big{(}|\Delta\psi(x,t)|^{2}+|\nabla\psi(x,t)|^{2}+|\psi(x,t)|^{2}-2w(x,t)\psi(x,t)\Big{)}\,dx\,dt.$

We see that $\phi$ is a weak solution of

$\begin{cases}\phi(x,t)-\Delta\phi(x,t)+\Delta^{2}\phi(x,t)=w(x,t)&{\mbox{ for
$(x,t)\in\Omega\times(0,T)$,}}\\\ \phi(x,t)=0&{\mbox{ for
$(x,t)\in(\partial\Omega)\times(0,T)$,}}\\\ \Delta\phi(x,t)=0&{\mbox{ for
$(x,t)\in(\partial\Omega)\times(0,T)$.}}\end{cases}$

Actually, $\phi(\cdot,t)\in H^{4}(\Omega)$, see e.g. Theorem 4 on page 317 of
[MR2597943], hence the equation is also fulfilled a.e., and thus, substituting
into (2.22),

$\int_{0}^{T}\langle w(\cdot,t),w(\cdot,t)\rangle_{L^{2}(\Omega)}\,dt=0.$

This gives that $w=0$, which in turn proves (2.21).

In particular, from (2.18), (2.19) and (2.21), if $w_{k}:=v_{k}-v$,

$\begin{split}&\sup_{k\in\mathds{N}}\|w_{k}\|_{L^{2}((0,T),\,H^{2}(\Omega))}=\sup_{k\in\mathds{N}}\|v_{k}-v\|_{L^{2}((0,T),\,H^{2}(\Omega))}=\sup_{k\in\mathds{N}}\|\sigma_{f_{u_{k}}}-\widetilde{v}\|_{L^{2}((0,T),\,H^{2}(\Omega))}\\\
&\qquad\leqslant\sup_{k\in\mathds{N}}\|\sigma_{f_{u_{k}}}\|_{L^{2}((0,T),\,H^{2}(\Omega))}+\|\widetilde{v}\|_{L^{2}((0,T),\,H^{2}(\Omega))}<+\infty.\end{split}$
(2.23)

Now we use the Gagliardo-Nirenberg Interpolation Inequality, see e.g. Theorem
1 in [MR208360] (used here with $a:=\frac{1}{2}$, $j:=1$, $m:=2$, $p:=2$,
$r:=2$, and $q:=2$), obtaining that

$\begin{split}&\int_{\Omega}|\nabla w_{k}(x,t)|^{2}\,dx\\\ &\qquad\leqslant
C\,\left[\left(\int_{\Omega}|D^{2}w_{k}(x,t)|^{2}\,dx\right)^{\frac{1}{2}}\left(\int_{\Omega}|w_{k}(x,t)|^{2}\,dx\right)^{\frac{1}{2}}+\int_{\Omega}|w_{k}(x,t)|^{2}\,dx\right],\end{split}$

for some $C>0$ depending only on $n$ and $\Omega$.

Hence, integrating for $t\in(0,T)$ and using the Hölder Inequality,

$\displaystyle\|w_{k}\|^{2}_{L^{2}((0,T),\,H^{1}(\Omega))}$
$\displaystyle\leqslant$ $\displaystyle
C\int_{0}^{T}\left(\int_{\Omega}|D^{2}w_{k}(x,t)|^{2}\,dx\right)^{\frac{1}{2}}\left(\int_{\Omega}|w_{k}(x,t)|^{2}\,dx\right)^{\frac{1}{2}}\,dt$
$\displaystyle\qquad\qquad\qquad+C\|w_{k}\|^{2}_{L^{2}((0,T),\,L^{2}(\Omega))}$
$\displaystyle\leqslant$ $\displaystyle
C\sqrt{\int_{0}^{T}\left(\int_{\Omega}|D^{2}w_{k}(x,t)|^{2}\,dx\right)\,dt\,\int_{0}^{T}\left(\int_{\Omega}|w_{k}(x,t)|^{2}\,dx\right)\,dt}$
$\displaystyle\qquad\qquad\qquad+C\|w_{k}\|^{2}_{L^{2}((0,T),\,L^{2}(\Omega))}$
$\displaystyle\leqslant$ $\displaystyle
C\Big{(}\|w_{k}\|_{L^{2}((0,T),\,H^{2}(\Omega))}\,\|w_{k}\|_{L^{2}((0,T),\,L^{2}(\Omega))}+\|w_{k}\|^{2}_{L^{2}((0,T),\,L^{2}(\Omega))}\Big{)}.$

From this, (2.20), and (2.23), we conclude that $w_{k}\to 0$ (and therefore
$v_{k}\to v$) in $X$. ∎

######  Corollary 2.10.

Let $\Omega$ be a bounded domain of $\mathds{R}^{n}$ with $C^{2}$ boundary.
Let also $K\in L^{2}(\Omega\times\Omega)$, $\Theta\in
L^{\infty}(\Omega\times(0,1))$, $\omega\in L^{\infty}(\Omega\times(0,1))$, and
$\beta\in W^{1,\infty}(\mathds{R})$.

Then, there exists $T_{\star}\in(0,1]$, depending only on $n$, $\Omega$,
$\|K\|_{L^{2}(\Omega\times\Omega)}$,
$\|\omega\|_{L^{\infty}(\Omega\times(0,1))}$, and
$\|\beta\|_{W^{1,\infty}(\mathds{R})}$, such that if $T\in(0,T_{\star}]$ the
following claims hold true.

The nonlinear operator $A$ satisfies $A:X\to X$.

Also, it is continuous and compact, and the set

$\big{\\{}{\mbox{$u\in X$ s.t. $u=\lambda Au$ for
some~{}$\lambda\in[0,1]$}}\big{\\}}$ (2.24)

is bounded.

###### Proof.

Let $T\in(0,1]$. By Lemma 2.8 we know that $A:X\to X$, and the continuity of
$A$ is a consequence of (2.16).

As for the compactness of $A$, let $u_{k}$ be a bounded sequence in $X$. Then,
by Lemma 2.7, the function $f_{u_{k}}$ is bounded in $Y$ and, consequently, by
(2.12),

$\sup_{k\in\mathds{N}}\|\sigma_{f_{u_{k}}}\|_{H^{1}((0,T),\,L^{2}(\Omega))}+\sup_{k\in\mathds{N}}\|\sigma_{f_{u_{k}}}\|_{L^{2}((0,T),\,H^{2}(\Omega))}<+\infty.$
(2.25)

On this account, using Lemma 2.9 with $v_{k}:=\sigma_{f_{u_{k}}}$, we see that
$v_{k}=\sigma_{f_{u_{k}}}=Av_{k}$ converges to $v$, thus establishing the
desired compactness property for the operator $A$.

Let now $u$ belong to the set in (2.24). Then, by (2.12) and Lemma 2.6,

$\displaystyle\|u\|_{X}=\lambda\|Au\|_{X}\leqslant\|Au\|_{X}=\|\sigma_{f_{u}}\|_{L^{2}((0,T),\,H^{1}(\Omega))}$
$\displaystyle\qquad=\sqrt{\int_{0}^{T}\|\sigma_{f_{u}}(\cdot,t)\|_{H^{1}(\Omega)}^{2}\,dt}\leqslant\sqrt{T\,\sup_{t\in[0,T]}\|\sigma_{f_{u}}(\cdot,t)\|_{H^{1}(\Omega)}^{2}}$
$\displaystyle\qquad\leqslant
C\sqrt{T}\,\Big{(}\|f_{u}\|_{Y}+\|g\|_{H^{1}(\Omega)}\Big{)},$

with $C>0$ depending only on $n$ and $\Omega$.

Hence, by Lemma 2.7,

$\displaystyle\|u\|_{X}\leqslant
C\sqrt{T}\,\left(\|u\|_{X}+\|u\|_{X}^{2-\gamma}+T\sup_{\Omega\times(0,T)}\Theta_{-}+\|g\|_{H^{1}(\Omega)}\right),$

with $C>0$ now depending only on $n$, $\Omega$,
$\|K\|_{L^{2}(\Omega\times\Omega)}$,
$\|\omega\|_{L^{\infty}(\Omega\times(0,T))}$, and
$\|\beta\|_{W^{1,\infty}(\mathds{R})}$.

Now we use Young Inequality with exponents $\frac{1}{2-\gamma}$ and
$\frac{1}{\gamma-1}$ and we see that

$\|u\|_{X}^{2-\gamma}\leqslant(2-\gamma)\|u\|_{X}+\gamma-1\leqslant
2\|u\|_{X}+1.$

Accordingly, up to renaming $C$,

$\displaystyle\|u\|_{X}\leqslant
C\sqrt{T}\,\left(\|u\|_{X}+T\sup_{\Omega\times(0,T)}\Theta_{-}+\|g\|_{H^{1}(\Omega)}+1\right),$

When $C\sqrt{T}\leqslant\frac{1}{2}$, we can reabsorb the term $\|u\|_{X}$ to
the left-hand side, thus obtaining, up to renaming constants,

$\|u\|_{X}\leqslant
C\left(\sup_{\Omega\times(0,T)}\Theta_{-}+\|g\|_{H^{1}(\Omega)}+1\right),$

proving that the set in (2.24) is bounded. ∎

With this preliminary work, we can now give the proof of Theorem 1.1 by
arguing as follows:

###### Proof of Theorem 1.1.

By Corollary 2.10, the hypotheses of Schaefer’s Fixed-Point Theorem are
satisfied (see e.g. Theorem 11.1 in [MR3967045]). This gives the existence of
a function $u$ satisfying $u=Au=\sigma_{f_{u}}$, that is

$\partial_{t}u=\Delta u+f_{u},$

yielding the desired result in view of (2.13). ∎

##  3\. Proof of Theorem 1.2

The gist of the proof of Theorem 1.2 consists in taking a sequence of
parameters $\gamma\searrow 1$ and the corresponding solutions $u_{\gamma}$
provided by Theorem 1.1, then utilizing the parabolic regularity theory to
infer uniform bounds and compactness properties that allow one to pass the
solutions to the limit.

The technical details go as follows:

###### Proof of Theorem 1.2.

We use the notation in (2.13), writing here $f^{(\gamma)}_{u}$ instead of
simply $f_{u}$ to make explicit the dependence on $\gamma$.

We let $T_{\star}$ be given by Theorem 1.1. Then, in the interval of time
$[0,T_{\star}]$, we obtain by Theorem 1.1 the existence of a solution
$u_{\gamma}$ of $\partial_{t}u_{\gamma}=\Delta
u_{\gamma}+f^{(\gamma)}_{u_{\gamma}}$, and, in view of (2.12),

$\begin{split}&\sup_{t\in[0,T_{\star}]}\|u_{\gamma}(\cdot,t)\|_{H^{1}(\Omega)}+\|u_{\gamma}\|_{L^{2}((0,T_{\star}),\,H^{2}(\Omega))}+\|\partial_{t}u_{\gamma}\|_{L^{2}((0,T_{\star}),\,L^{2}(\Omega))}\\\
&\qquad\leqslant
C\Big{(}\|f^{(\gamma)}_{u_{\gamma}}\|_{L^{2}((0,T_{\star}),\,L^{2}(\Omega))}+\|g\|_{H^{1}(\Omega)}\Big{)},\end{split}$
(3.1)

for some $C>0$ depending only on $n$ and $\Omega$ (recall Lemma 2.6).

Furthermore, in light of Lemmata 2.2 and 2.3, setting ${u^{(t)}}:=u(\cdot,t)$,
we have that

$\begin{split}\|f^{(\gamma)}_{u_{\gamma}}(\cdot,t)\|_{L^{2}(\Omega)}\leqslant\,&\sqrt{2}\|K\|_{L^{2}(\Omega\times\Omega)}\,\left(\|u^{(t)}_{\gamma}\|_{L^{2}(\Omega)}+|\Omega|^{\frac{1}{2}}\sup_{\Omega\times(0,T_{\star})}\Theta_{-}\right)\\\
&\quad+\sqrt{2}\left(\|\omega(\cdot,t)\|_{L^{\infty}(\Omega)}\,\|\nabla
u^{(t)}_{\gamma}\|_{L^{2}(\Omega)}+\|\beta\|_{L^{\infty}(\mathds{R})}\,|\Omega|^{\frac{\gamma-1}{2}}\|\nabla
u^{(t)}_{\gamma}\|_{L^{2}(\Omega)}^{2-\gamma}\right)\\\
\leqslant\,&C\Big{(}\|K\|_{L^{2}(\Omega\times\Omega)}+\|\omega(\cdot,t)\|_{L^{\infty}(\Omega)}+\|\beta\|_{L^{\infty}(\mathds{R})}\Big{)}\Big{(}\|u^{(t)}_{\gamma}\|_{H^{1}(\Omega)}+1\Big{)}\\\
\leqslant\,&C\epsilon_{0}\Big{(}\|u^{(t)}_{\gamma}\|_{H^{1}(\Omega)}+1\Big{)},\end{split}$

up to renaming $C>0$, possibly in dependence of $n$, $\Omega$ and
$\|\Theta_{-}\|_{L^{\infty}(\Omega\times(0,1))}$.

As a result,

$\|f^{(\gamma)}_{u_{\gamma}}\|_{L^{2}((0,T_{\star}),\,L^{2}(\Omega))}\leqslant
C\epsilon_{0}\Big{(}\|u_{\gamma}\|_{L^{2}((0,T_{\star}),\,H^{1}(\Omega))}+1\Big{)},$

up to renaming $C$, which, combined with (3.1), gives that

$\begin{split}&\sup_{t\in[0,T_{\star}]}\|u_{\gamma}(\cdot,t)\|_{H^{1}(\Omega)}+\|u_{\gamma}\|_{L^{2}((0,T_{\star}),\,H^{2}(\Omega))}+\|\partial_{t}u_{\gamma}\|_{L^{2}((0,T_{\star}),\,L^{2}(\Omega))}\\\
&\qquad\leqslant
C\Big{(}\epsilon_{0}\|u_{\gamma}\|_{L^{2}((0,T_{\star}),\,H^{1}(\Omega))}+1+\|g\|_{H^{1}(\Omega)}\Big{)}.\end{split}$

Consequently, if $\epsilon_{0}$ is small enough (possibly in dependence of
$n$, $\Omega$ and $\|\Theta_{-}\|_{L^{\infty}(\Omega\times(0,1))}$), we can
reabsorb one term to the left-hand side and conclude that

$\begin{split}&\sup_{t\in[0,T_{\star}]}\|u_{\gamma}(\cdot,t)\|_{H^{1}(\Omega)}+\|u_{\gamma}\|_{L^{2}((0,T_{\star}),\,H^{2}(\Omega))}+\|\partial_{t}u_{\gamma}\|_{L^{2}((0,T_{\star}),\,L^{2}(\Omega))}\\\
&\qquad\qquad\leqslant C\Big{(}1+\|g\|_{H^{1}(\Omega)}\Big{)}.\end{split}$
(3.2)

We now take a sequence $\gamma_{k}\searrow 1$ and we obtain from (3.2) by
compactness (see Lemma 2.9) that

$u_{\gamma_{k}}$ converges, up to a subsequence, to some $u$ in
$L^{2}((0,T_{\star}),\,H^{1}(\Omega))$. (3.3)

Another consequence of (3.2) is that

$\iint_{\Omega\times(0,T_{\star})}|\partial_{t}u_{\gamma_{k}}(x,t)|^{2}\,dx\,dt\leqslant
C^{2}\Big{(}1+\|g\|_{H^{1}(\Omega)}\Big{)}^{2}$

and so, up to a subsequence,

$\partial_{t}u_{\gamma_{k}}$ converges weakly to some function $U$ in
$L^{2}(\Omega\times(0,T_{\star}))$. (3.4)

We claim that, in the weak sense,

$U=\partial_{t}u.$ (3.5)

Indeed, for all $\psi\in C^{\infty}_{0}(\Omega\times(0,T_{\star}))$, we infer
from (3.3) and (3.4) that

$\displaystyle\iint_{\Omega\times(0,T_{\star})}U(x,t)\,\psi(x,t)\,dx\,dt=\lim_{k\to+\infty}\iint_{\Omega\times(0,T_{\star})}\partial_{t}u_{\gamma_{k}}(x,t)\,\psi(x,t)\,dx\,dt$
$\displaystyle\qquad=-\lim_{k\to+\infty}\iint_{\Omega\times(0,T_{\star})}u_{\gamma_{k}}(x,t)\,\partial_{t}\psi(x,t)\,dx\,dt=-\iint_{\Omega\times(0,T_{\star})}u(x,t)\,\partial_{t}\psi(x,t)\,dx\,dt,$

which establishes (3.5).

It remains to show that $\partial_{t}u=\Delta u+f^{(1)}_{u}$ in the weak
sense, i.e. (see [MR2597943, page 352]) that, for a.e. $t\in[0,T_{\star}]$ and
all $\psi\in H^{1}_{0}(\Omega)$,

$\langle\partial_{t}u(\cdot,t),\psi\rangle_{H^{-1}(\Omega),\,H^{1}_{0}(\Omega)}+\int_{\Omega}\nabla
u(x,t)\cdot\nabla\psi(x)\,dx-\int_{\Omega}f^{(1)}_{u}(x,t)\,\psi(x)\,dx=0.$
(3.6)

To this end, we let $w_{k}:=u-u_{\gamma_{k}}$ and, in light of (3.3), we
recall that

$\lim_{k\to+\infty}\|w_{k}\|_{L^{2}((0,T_{\star}),\,H^{1}(\Omega))}=0.$ (3.7)

Moreover, by (3.4) and (3.5), as $k\to+\infty$,

$\partial_{t}w_{k}\to 0$ weakly in $L^{2}(\Omega\times(0,T_{\star}))$. (3.8)

We also call $\zeta(t)$ the left-hand side of (3.6) (in this way, to prove
(3.6), we will need to show that $\zeta$ vanishes almost everywhere).

Since

$\langle\partial_{t}u_{\gamma_{k}}(\cdot,t),\psi\rangle_{H^{-1}(\Omega),\,H^{1}_{0}(\Omega)}+\int_{\Omega}\nabla
u_{\gamma_{k}}(x,t)\cdot\nabla\psi(x)\,dx-\int_{\Omega}f^{(\gamma_{k})}_{u_{\gamma_{k}}}(x,t)\,\psi(x)\,dx=0,$

we find that

$\begin{split}&\zeta(t)=\langle\partial_{t}w_{k}(\cdot,t),\psi\rangle_{H^{-1}(\Omega),\,H^{1}_{0}(\Omega)}+\int_{\Omega}\nabla
w_{k}(x,t)\cdot\nabla\psi(x)\,dx\\\
&\qquad\qquad\qquad-\int_{\Omega}\Big{(}f^{(1)}_{u}(x,t)-f^{(\gamma_{k})}_{u_{\gamma_{k}}}(x,t)\Big{)}\,\psi(x)\,dx.\end{split}$
(3.9)

Let now make the duality
$\langle\cdot,\cdot\rangle_{H^{-1}(\Omega),\,H^{1}_{0}(\Omega)}$ more explicit
by using the canonical embedding of $L^{2}(\Omega)$ into $H^{-1}(\Omega)$.
Specifically, one identifies $L^{2}(\Omega)$ with its dual, which is contained
in the dual of $H^{1}_{0}(\Omega)$, which is identified with $H^{-1}(\Omega)$,
therefore, for $h\in L^{2}(\Omega)\subset H^{-1}(\Omega)$, the pairing
$\langle h,\psi\rangle_{H^{-1}(\Omega),\,H^{1}_{0}(\Omega)}$ reduces to
$\int_{\Omega}h(x)\,\psi(x)\,dx$, which is indeed a bounded linear functional
on $H^{1}_{0}(\Omega)$.

Consequently, for every $\Psi\in C^{\infty}_{0}((0,T_{\star}))$, we deduce
from (3.8) that

$\lim_{k\to+\infty}\int_{0}^{T_{\star}}\langle\partial_{t}w_{k}(\cdot,t),\psi\rangle_{H^{-1}(\Omega),\,H^{1}_{0}(\Omega)}\,\Psi(t)\,dt=\lim_{k\to+\infty}\iint_{\Omega\times(0,T_{\star})}\partial_{t}w_{k}(x,t)\,\psi(x)\,\Psi(t)\,dx\,dt=0.$

Plugging this information into (3.9) we conclude that, for every $\Psi\in
C^{\infty}_{0}((0,T_{\star}))$,

$\begin{split}\int_{0}^{T_{\star}}\zeta(t)\,\Psi(t)\,dt&=\lim_{k\to+\infty}\Bigg{(}\,\iint_{\Omega\times(0,T_{\star})}\nabla
w_{k}(x,t)\cdot\nabla\psi(x)\,\Psi(t)\,dx\,dt\\\
&\qquad\qquad-\iint_{\Omega\times(0,T_{\star})}\Big{(}f^{(1)}_{u}(x,t)-f^{(\gamma_{k})}_{u_{\gamma_{k}}}(x,t)\Big{)}\,\psi(x)\,\Psi(t)\,dx\,dt\Bigg{)}.\end{split}$
(3.10)

Also,

$\displaystyle\left|\,\iint_{\Omega\times(0,T_{\star})}\nabla
w_{k}(x,t)\cdot\nabla\psi(x)\,\Psi(t)\,dx\,dt\right|^{2}$
$\displaystyle\qquad\leqslant
T_{\star}\,|\Omega|\iint_{\Omega\times(0,T_{\star})}|\nabla
w_{k}(x,t)|^{2}\,|\nabla\psi(x)|^{2}\,|\Psi(t)|^{2}\,dx\,dt$
$\displaystyle\qquad\leqslant
T_{\star}\,|\Omega|\,\|\nabla\psi\|^{2}_{L^{\infty}(\Omega)}\,\|\Psi\|^{2}_{L^{\infty}((0,T_{\star}))}\,\|w_{k}\|_{L^{2}((0,T_{\star}),\,H^{1}(\Omega))}^{2},$

which, owing to (3.7), is infinitesimal as $k\to+\infty$.

As a consequence, (3.10) boils down to

$\begin{split}&\int_{0}^{T_{\star}}\zeta(t)\,\Psi(t)\,dt=-\lim_{k\to+\infty}\iint_{\Omega\times(0,T_{\star})}\Big{(}f^{(1)}_{u}(x,t)-f^{(\gamma_{k})}_{u_{\gamma_{k}}}(x,t)\Big{)}\,\psi(x)\,\Psi(t)\,dx\,dt.\end{split}$
(3.11)

Now we use the notation in (2.3). We also adopt the notation in (2.4), but
writing $f_{2,u}^{(\gamma)}$ in lieu of $f_{2,u}$ to emphasize the dependence
on $\gamma$. In this way, we can write that

$f^{(\gamma)}_{u}=f_{1,u}+f_{2,u}^{(\gamma)}.$ (3.12)

By Lemma 2.4, we know that

$\begin{split}\|f_{1,u}(\cdot,t)-f_{1,{u_{\gamma_{k}}}}(\cdot,t)\|_{L^{2}(\Omega)}&\leqslant\|K\|_{L^{2}(\Omega\times\Omega)}\|u(\cdot,t)-{u_{\gamma_{k}}}(\cdot,t)\|_{L^{2}(\Omega)}\\\
&=\|K\|_{L^{2}(\Omega\times\Omega)}\|w_{k}(\cdot,t)\|_{L^{2}(\Omega)}\end{split}$

and therefore

$\begin{split}&\left|\,\iint_{\Omega\times(0,T_{\star})}\Big{(}f_{1,u}(x,t)-f_{1,u_{\gamma_{k}}}(x,t)\Big{)}\,\psi(x)\,\Psi(t)\,dx\,dt\right|^{2}\\\
&\qquad\leqslant
T_{\star}\,|\Omega|\,\|\nabla\psi\|^{2}_{L^{\infty}(\Omega)}\,\|\Psi\|^{2}_{L^{\infty}((0,T_{\star}))}\iint_{\Omega\times(0,T_{\star})}\Big{(}f_{1,u}(x,t)-f_{1,u_{\gamma_{k}}}(x,t)\Big{)}^{2}\,dx\,dt\\\
&\qquad=T_{\star}\,|\Omega|\,\|\nabla\psi\|^{2}_{L^{\infty}(\Omega)}\,\|\Psi\|^{2}_{L^{\infty}((0,T_{\star}))}\int_{0}^{T_{\star}}\|f_{1,u}(\cdot,t)-f_{1,{u_{\gamma_{k}}}}(\cdot,t)\|_{L^{2}(\Omega)}^{2}\,dt\\\
&\qquad\leqslant
T_{\star}\,|\Omega|\,\|\nabla\psi\|^{2}_{L^{\infty}(\Omega)}\,\|\Psi\|^{2}_{L^{\infty}((0,T_{\star}))}\|K\|_{L^{2}(\Omega\times\Omega)}^{2}\int_{0}^{T_{\star}}\|w_{k}(\cdot,t)\|^{2}_{L^{2}(\Omega)}\,dt\\\
&\qquad=T_{\star}\,|\Omega|\,\|\nabla\psi\|^{2}_{L^{\infty}(\Omega)}\,\|\Psi\|^{2}_{L^{\infty}((0,T_{\star}))}\|K\|_{L^{2}(\Omega\times\Omega)}^{2}\,\|w_{k}\|_{L^{2}((0,T_{\star}),\,L^{2}(\Omega))}^{2},\end{split}$

which is infinitesimal as $k\to+\infty$, thanks to (3.7).

By virtue of this observation and (3.12), we can reduce (3.11) to

$\begin{split}&\int_{0}^{T_{\star}}\zeta(t)\,\Psi(t)\,dt=-\lim_{k\to+\infty}\iint_{\Omega\times(0,T_{\star})}\Big{(}f^{(1)}_{2,u}(x,t)-f^{(\gamma_{k})}_{2,u_{\gamma_{k}}}(x,t)\Big{)}\,\psi(x)\,\Psi(t)\,dx\,dt.\end{split}$
(3.13)

Now we remark that

$\begin{split}&\Big{|}f^{(1)}_{2,u}(x,t)-f^{(\gamma_{k})}_{2,u_{\gamma_{k}}}(x,t)\Big{|}\\\
=&\left|\left(\left(\omega(x,t)+\frac{\beta(u(x,t))\nabla u(x,t)}{|\nabla
u(x,t)|}\right)\cdot\nabla
u(x,t)\right)_{-}-\left(\left(\omega(x,t)+\frac{\beta(u_{\gamma_{k}}(x,t))\nabla
u_{\gamma_{k}}(x,t)}{|\nabla
u_{\gamma_{k}}(x,t)|^{\gamma_{k}}}\right)\cdot\nabla
u_{\gamma_{k}}(x,t)\right)_{-}\right|\\\
\leqslant&\left|\left(\omega(x,t)+\frac{\beta(u(x,t))\nabla u(x,t)}{|\nabla
u(x,t)|}\right)\cdot\nabla
u(x,t)-\left(\omega(x,t)+\frac{\beta(u_{\gamma_{k}}(x,t))\nabla
u_{\gamma_{k}}(x,t)}{|\nabla
u_{\gamma_{k}}(x,t)|^{\gamma_{k}}}\right)\cdot\nabla
u_{\gamma_{k}}(x,t)\right|\\\
\leqslant&\|\omega\|_{L^{\infty}(\Omega\times(0,1))}\big{|}\nabla
u(x,t)-\nabla u_{\gamma_{k}}(x,t)\big{|}+\Big{|}\beta(u(x,t))|\nabla
u(x,t)|-\beta(u_{\gamma_{k}}(x,t))|\nabla
u_{\gamma_{k}}(x,t)|^{2-\gamma_{k}}\Big{|}\\\
\leqslant&\|\omega\|_{L^{\infty}(\Omega\times(0,1))}\big{|}\nabla
w_{k}(x,t)\big{|}+\|\beta\|_{L^{\infty}(\mathds{R})}\Big{|}|\nabla
u(x,t)|-|\nabla u_{\gamma_{k}}(x,t)|^{2-\gamma_{k}}\Big{|}\\\
&\quad+\big{|}\beta(u(x,t))-\beta(u_{\gamma_{k}}(x,t))\big{|}\,|\nabla
u_{\gamma_{k}}(x,t)|^{2-\gamma_{k}}\\\
\leqslant&\|\omega\|_{L^{\infty}(\Omega\times(0,1))}\big{|}\nabla
w_{k}(x,t)\big{|}+\|\beta\|_{L^{\infty}(\mathds{R})}\Big{|}|\nabla
u(x,t)|-|\nabla u(x,t)|^{2-\gamma_{k}}\Big{|}\\\
&\quad+\|\beta\|_{L^{\infty}(\mathds{R})}\Big{|}|\nabla
u(x,t)|^{2-\gamma_{k}}-|\nabla
u_{\gamma_{k}}(x,t)|^{2-\gamma_{k}}\Big{|}+\sqrt{2}\|\beta\|_{W^{1,\infty}(\mathds{R})}|u(x,t))-u_{\gamma_{k}}(x,t)|^{\frac{1}{2}}\,|\nabla
u_{\gamma_{k}}(x,t)|^{2-\gamma_{k}}.\end{split}$

Hence, by (2.10),

$\displaystyle\Big{|}f^{(1)}_{2,u}(x,t)-f^{(\gamma_{k})}_{2,u_{\gamma_{k}}}(x,t)\Big{|}$
$\displaystyle\leqslant$ $\displaystyle C\Big{(}\big{|}\nabla
w_{k}(x,t)\big{|}+\big{|}|\nabla u(x,t)|-|\nabla
u(x,t)|^{2-\gamma_{k}}\big{|}+\big{|}\nabla
w_{k}(x,t)\big{|}^{2-\gamma_{k}}+|w_{k}(x,t)|^{\frac{1}{2}}\,|\nabla
u_{\gamma_{k}}(x,t)|^{2-\gamma_{k}}\Big{)},$

for some $C>0$.

As a result,

$\displaystyle\iint_{\Omega\times(0,T_{\star})}\Big{|}f^{(1)}_{2,u}(x,t)-f^{(\gamma_{k})}_{2,u_{\gamma_{k}}}(x,t)\Big{|}^{2}\,dx\,dt$
$\displaystyle\leqslant$ $\displaystyle
C\Bigg{(}\|w_{k}\|_{L^{2}((0,T_{\star}),\,H^{1}(\Omega))}^{2}+\iint_{\Omega\times(0,T_{\star})}\big{|}|\nabla
u(x,t)|-|\nabla u(x,t)|^{2-\gamma_{k}}\big{|}^{2}\,dx\,dt$
$\displaystyle\qquad+\|w_{k}\|_{L^{2}((0,T_{\star}),\,H^{1}(\Omega))}^{2(2-\gamma_{k})}+\|w_{k}\|_{L^{2}((0,T_{\star}),\,L^{2}(\Omega))}\|\nabla
u_{\gamma_{k}}\|^{2(2-\gamma_{k})}_{L^{2}((0,T_{\star}),\,L^{2}(\Omega))}\Bigg{)},$

which is infinitesimal as $k\to+\infty$, thanks to (3.2), (3.7), and the
Dominated Convergence Theorem.

From this observation and (3.13) we conclude that

$\int_{0}^{T_{\star}}\zeta(t)\,\Psi(t)\,dt=0.$

Since $\Psi\in C^{\infty}_{0}((0,T_{\star}))$ is arbitrary, we conclude that
$\zeta=0$ almost everywhere in $[0,T_{\star}]$, as desired. ∎

##  4\. Proof of Theorem 1.3

###  4.1. Toolbox

The gist to prove the global existence result in Theorem 1.3 is that the local
existence result in Theorem 1.1 will provide a solution defined in a uniform
interval of time length $T_{\star}$, only depending on $n$, $\Omega$,
$\|K\|_{L^{2}(\Omega\times\Omega)}$,
$\|\omega\|_{L^{\infty}(\Omega\times(0,1))}$, and
$\|\beta\|_{W^{1,\infty}(\mathds{R})}$, therefore one can iterate Theorem 1.1,
starting the flow at $u(\cdot,T_{\star})$, thus obtaining a solution up to
time $2T_{\star}$, and so on.

The details to make this idea work are perhaps a bit technical, since the
equation under consideration is not standard, therefore, for completeness, we
provide the complete argument.

This section collects some ancillary results towards the proof of Theorem 1.3.
The following is a continuity property of the solution at a given time.

######  Lemma 4.1.

Let $T_{\star}$ be as in Theorem 1.1. Let $T\in(0,T_{\star}]$ and $u$ be given
by Theorem 1.1.

Then,

$\begin{split}&\sup_{t\in[0,T]}\|u(\cdot,t)\|_{H^{1}(\Omega)}+\|u\|_{L^{2}((0,T),\,H^{2}(\Omega))}+\|\partial_{t}u\|_{L^{2}((0,T),\,L^{2}(\Omega))}\\\
&\qquad\leqslant C\Big{(}1+\|u(\cdot,0)\|_{H^{1}(\Omega)}\Big{)},\end{split}$
(4.1)

for some $C>0$ depending only on $n$, $\Omega$,
$\|K\|_{L^{2}(\Omega\times\Omega)}$,
$\|\omega\|_{L^{\infty}(\Omega\times(0,T))}$,
$\|\beta\|_{W^{1,\infty}(\mathds{R})}$, and
$\|\Theta_{-}\|_{L^{\infty}(\Omega\times(0,T))}$.

Moreover, for all $\widetilde{T}\in(0,T]$,

$\lim_{\eta\searrow
0}\|u(\cdot,\widetilde{T}-\eta)-u(\cdot,\widetilde{T})\|_{L^{2}(\Omega)}=0$
(4.2)

and, for all $\widetilde{T}\in[0,T)$,

$\lim_{\eta\searrow
0}\|u(\cdot,\widetilde{T}+\eta)-u(\cdot,\widetilde{T})\|_{L^{2}(\Omega)}=0.$
(4.3)

###### Proof.

In the notation of (2.13), we deduce from (2.12) that

$\begin{split}&\sup_{t\in[0,T]}\|u(\cdot,t)\|_{H^{1}(\Omega)}+\|u\|_{L^{2}((0,T),\,H^{2}(\Omega))}+\|\partial_{t}u\|_{L^{2}((0,T),\,L^{2}(\Omega))}\\\
&\qquad\leqslant
C\Big{(}\|f_{u}\|_{L^{2}((0,T),\,L^{2}(\Omega))}+\|u(\cdot,0)\|_{H^{1}(\Omega)}\Big{)},\end{split}$
(4.4)

with $C>0$ depending only on $n$ and $\Omega$ (recall Lemma 2.6).

Moreover, since the set in (2.24) is bounded,

$\|u\|_{L^{2}((0,T),\,H^{1}_{0}(\Omega))}\leqslant
C\left(\sup_{\Omega\times(0,T)}\Theta_{-}+\|u(\cdot,0)\|_{H^{1}(\Omega)}+1\right),$
(4.5)

with $C>0$ now depending also on $\|K\|_{L^{2}(\Omega\times\Omega)}$,
$\|\omega\|_{L^{\infty}(\Omega\times(0,T))}$, and
$\|\beta\|_{W^{1,\infty}(\mathds{R})}$.

Furthermore, in light of Lemma 2.7,

$\|f_{u}\|_{L^{2}((0,T),\,L^{2}(\Omega))}\leqslant
C\,\left(\|u\|_{L^{2}((0,T),\,H^{1}_{0}(\Omega))}+\sup_{\Omega\times(0,T)}\Theta_{-}\right).$

Combining this information and (4.5) we obtain that

$\|f_{u}\|_{L^{2}((0,T),\,L^{2}(\Omega))}\leqslant
C\Big{(}1+\|u(\cdot,0)\|_{H^{1}(\Omega)}\Big{)},$

up to renaming $C$ that now may depend also on
$\|\Theta_{-}\|_{L^{\infty}(\Omega\times(0,T))}$.

This in turn, together with (4.4), gives (4.1), as desired.

Now we prove (4.2). For this, we observe that, for all $t\in(0,\eta)$,

$\displaystyle|u(x,\widetilde{T}-\eta)-u(x,\widetilde{T})|^{2}\leqslant\left(\int_{0}^{\eta}|\partial_{t}u(x,\widetilde{T}-s)|\,ds\right)^{2}\leqslant\eta\int_{0}^{\eta}|\partial_{t}u(x,\widetilde{T}-s)|^{2}\,ds.$

Hence, integrating over $x\in\Omega$,

$\displaystyle\|u(\cdot,\widetilde{T}-\eta)-u(\cdot,\widetilde{T})\|^{2}_{L^{2}(\Omega)}\leqslant\eta\int_{0}^{\eta}\|\partial_{t}u(\cdot,\widetilde{T}-s)\|^{2}_{L^{2}(\Omega)}\,ds.$

This and (4.1) give that

$\displaystyle\|u(\cdot,\widetilde{T}-t)-u(\cdot,t)\|^{2}_{L^{2}(\Omega)}\leqslant
C\eta\Big{(}1+\|u(\cdot,0)\|_{H^{1}(\Omega)}\Big{)},$

from which (4.2) follows.

The proof of (4.3) is analogous to that of (4.2). ∎

The following result is the core step to iterate the local result in Theorem
1.1 to obtain a global solution in Theorem 1.3. The idea is to “glue together”
two solutions of

$\begin{cases}\partial_{t}u(x,\cdot)=\Delta
u(x,\cdot)+\displaystyle\int_{\Omega}\big{(}u(y,\cdot)-\Theta(y,\cdot)\big{)}_{+}\,K(x,y)\,dy\\\
\qquad\qquad\qquad\qquad+\displaystyle\left(\left(\omega+\frac{\beta(u)\nabla
u}{|\nabla u|^{\gamma}}\right)\cdot\nabla u\right)_{-}\\\ \qquad{\mbox{ for
$x\in\Omega$,}}\\\ u(x,\cdot)=0\\\ \qquad{\mbox{ for
$x\in\partial\Omega$,}}\end{cases}$ (4.6)

whenever the final value of the first solution coincides with the initial
value of the second solution.

For this, we let $T_{\star}$ be as in Theorem 1.1 and $\widetilde{u}_{0}$ be a
solution of (4.6) in $[0,T_{\star}]$ with initial datum $g_{0}$, as given by
Theorem 1.1.

We now argue recursively to construct a global solution. For all
$k\in\mathds{N}$ with $k\geqslant 1$, we set
$g_{k}:=\widetilde{u}_{k-1}(\cdot,kT_{\star})$ and let $\widetilde{u}_{k}$ be
a solution of (4.6) in the time interval $[kT_{\star},(k+1)T_{\star}]$ with
initial datum $g_{k}$, as given by Theorem 1.1.

Now let $u_{0}:=\widetilde{u}_{0}$ and, for all $k\in\mathds{N}$ with
$k\geqslant 1$,

$u_{k}(\cdot,t):=\begin{cases}u_{k-1}(\cdot,t)&{\mbox{ if
}}t\in[0,kT_{\star}],\\\ \widetilde{u}_{k}(\cdot,t)&{\mbox{ if
}}t\in(kT_{\star},(k+1)T_{\star}].\end{cases}$ (4.7)

With this notation, we have the following:

######  Lemma 4.2.

For all $k\in\mathds{N}$, we have that $u_{k}$ is a solution of (4.6) in the
time interval $[0,(k+1)T_{\star}]$.

###### Proof.

When $k=0$, we have that $u_{0}=\widetilde{u}_{0}$ is a solution of (4.6) in
$[0,T_{\star}]$, as desired.

When $k=1$, we have that

$u_{1}(\cdot,t):=\begin{cases}u_{0}(\cdot,t)&{\mbox{ if
}}t\in[0,T_{\star}],\\\ \widetilde{u}_{1}(\cdot,t)&{\mbox{ if
}}t\in(T_{\star},2T_{\star}],\end{cases}$

and we claim that

$u_{1}$ is a solution of (4.6) in the time interval $[0,2T_{\star}]$. (4.8)

To establish this, we let $\eta>0$ (to be taken as small as we wish here
below) and $\tau_{\eta}\in C^{\infty}(\mathds{R})$ such that
$\tau_{\eta}(T_{\star}-t)=\tau_{\eta}(T_{\star}+t)$ for all $t\in\mathds{R}$,
$\tau_{\eta}=0$ in $(T_{\star}-\eta,T_{\star}+\eta)$, $\tau_{\eta}=1$ in
$\mathds{R}\setminus(T_{\star}-2\eta,T_{\star}+2\eta)$, and
$\|\tau_{\eta}\|_{L^{\infty}(\mathds{R})}\leqslant\frac{2}{\eta}$. Given
$\phi\in C^{\infty}_{0}((0,2T_{\star}))$, we let
$\phi_{\eta}:=\tau_{\eta}\phi$ and we remark that

$\begin{split}&\left|\int_{0}^{2T_{\star}}\phi^{\prime}(t)\,u_{1}(x,t)\,dt-\int_{0}^{2T_{\star}}\phi^{\prime}_{\eta}(t)\,u_{1}(x,t)\,dt\right|\\\
&\qquad=\left|\int_{0}^{2T_{\star}}\Big{(}\phi^{\prime}(t)\big{(}1-\tau_{\eta}(t)\big{)}-\phi(t)\tau_{\eta}^{\prime}(t)\Big{)}\,u_{1}(x,t)\,dt\right|\\\
&\qquad\leqslant\|\phi^{\prime}\|_{L^{\infty}(\mathds{R})}\int_{T_{\star}-2\eta}^{T_{\star}+2\eta}|u_{1}(x,t)|\,dt\\\
&\qquad\qquad+\left|\int_{T_{\star}-2\eta}^{T_{\star}}\phi(t)\tau_{\eta}^{\prime}(t)u_{0}(x,t)\,dt+\int_{T_{\star}}^{T_{\star}+2\eta}\phi(t)\tau_{\eta}^{\prime}(t)\widetilde{u}_{1}(x,t)\,dt\right|\\\
&\qquad\leqslant
2\|\phi^{\prime}\|_{L^{\infty}(\mathds{R})}\sqrt{\eta}\,\sqrt{\int_{T_{\star}-2\eta}^{T_{\star}+2\eta}|u_{1}(x,t)|^{2}\,dt}\\\
&\qquad\qquad+\left|\int_{0}^{2\eta}\phi(T_{\star}-s)\tau_{\eta}^{\prime}(T_{\star}-s)u_{0}(x,T_{\star}-s)\,ds\right.\\\
&\qquad\qquad\qquad\left.+\int_{0}^{2\eta}\phi(T_{\star}+s)\tau_{\eta}^{\prime}(T_{\star}+s)\widetilde{u}_{1}(x,T_{\star}+s)\,ds\right|\\\
&\qquad\leqslant
2\|\phi^{\prime}\|_{L^{\infty}(\mathds{R})}\sqrt{\eta}\,\sqrt{\int_{T_{\star}-2\eta}^{T_{\star}+2\eta}|u_{1}(x,t)|^{2}\,dt}\\\
&\qquad\qquad+\frac{2}{\eta}\int_{0}^{2\eta}\Big{|}\phi(T_{\star}-s)u_{0}(x,T_{\star}-s)-\phi(T_{\star}+s)\widetilde{u}_{1}(x,T_{\star}+s)\Big{|}\,ds\\\
&\qquad\leqslant
2\|\phi^{\prime}\|_{L^{\infty}(\mathds{R})}\sqrt{\eta}\,\sqrt{\int_{T_{\star}-2\eta}^{T_{\star}+2\eta}|u_{1}(x,t)|^{2}\,dt}\\\
&\qquad\qquad+\frac{2}{\eta}\int_{0}^{2\eta}\Big{|}\big{(}\phi(T_{\star}-s)-\phi(T_{\star}+s)\big{)}u_{0}(x,T_{\star}-s)\Big{|}\,ds\\\
&\qquad\qquad+\frac{2}{\eta}\int_{0}^{2\eta}\Big{|}\phi(T_{\star}+s)\big{(}\widetilde{u}_{1}(x,T_{\star}-s)-\widetilde{u}_{1}(x,T_{\star}+s)\big{)}\Big{|}\,ds\\\
&\qquad\leqslant
2\|\phi^{\prime}\|_{L^{\infty}(\mathds{R})}\sqrt{\eta}\,\sqrt{\int_{T_{\star}-2\eta}^{T_{\star}+2\eta}|u_{1}(x,t)|^{2}\,dt}+8\|\phi^{\prime}\|_{L^{\infty}(\mathds{R})}\int_{0}^{2\eta}|u_{0}(x,T_{\star}-s)|\,ds\\\
&\qquad\qquad+\frac{2\|\phi\|_{L^{\infty}(\mathds{R})}}{\eta}\int_{0}^{2\eta}\big{|}u_{0}(x,T_{\star}-s)-\widetilde{u}_{1}(x,T_{\star}+s)\big{|}\,ds\\\
&\qquad\leqslant
C\sqrt{\eta\int_{T_{\star}-2\eta}^{T_{\star}+2\eta}|u_{1}(x,t)|^{2}\,dt+\frac{1}{\eta}\int_{0}^{2\eta}\big{|}u_{0}(x,T_{\star}-s)-\widetilde{u}_{1}(x,T_{\star}+s)\big{|}^{2}\,ds},\end{split}$

for some $C>0$.

As a result, up to renaming $C$ line after line,

$\begin{split}&\left\|\int_{0}^{2T_{\star}}\phi^{\prime}(t)\,u_{1}(\cdot,t)\,dt-\int_{0}^{2T_{\star}}\phi^{\prime}_{\eta}(t)\,u_{1}(\cdot,t)\,dt\right\|_{L^{2}(\Omega)}^{2}\\\
&\qquad\leqslant
C\int_{\Omega}\left(\eta\int_{T_{\star}-2\eta}^{T_{\star}+2\eta}|u_{1}(x,t)|^{2}\,dt+\frac{1}{\eta}\int_{0}^{2\eta}\big{|}u_{0}(x,T_{\star}-s)-\widetilde{u}_{1}(x,T_{\star}+s)\big{|}^{2}\,ds\right)\,dx\\\
&\qquad=C\left(\eta\int_{T_{\star}-2\eta}^{T_{\star}+2\eta}\|u_{1}(\cdot,t)\|^{2}_{L^{2}(\Omega)}\,dt+\frac{1}{\eta}\int_{0}^{2\eta}\|u_{0}(\cdot,T_{\star}-s)-\widetilde{u}_{1}(\cdot,T_{\star}+s)\|^{2}_{L^{2}(\Omega)}\,ds\right)\end{split}$

and therefore

$\begin{split}&\lim_{\eta\searrow
0}\left\|\int_{0}^{2T_{\star}}\phi^{\prime}(t)\,u_{1}(\cdot,t)\,dt-\int_{0}^{2T_{\star}}\phi^{\prime}_{\eta}(t)\,u_{1}(\cdot,t)\,dt\right\|_{L^{2}(\Omega)}^{2}\\\
&\qquad\leqslant C\lim_{\eta\searrow
0}\|u_{0}(\cdot,T_{\star}-2\eta)-\widetilde{u}_{1}(\cdot,T_{\star}+2\eta)\|^{2}_{L^{2}(\Omega)}\\\
&\qquad\leqslant C\lim_{\eta\searrow
0}\Big{(}\|u_{0}(\cdot,T_{\star}-2\eta)-u_{0}(\cdot,T_{\star})\|^{2}_{L^{2}(\Omega)}+\|\widetilde{u}_{1}(\cdot,T_{\star})-\widetilde{u}_{1}(\cdot,T_{\star}+2\eta)\|^{2}_{L^{2}(\Omega)}\Big{)}.\end{split}$

From this, (4.2) and (4.3), it follows that

$\lim_{\eta\searrow
0}\left\|\int_{0}^{2T_{\star}}\phi^{\prime}(t)\,u_{1}(\cdot,t)\,dt-\int_{0}^{2T_{\star}}\phi^{\prime}_{\eta}(t)\,u_{1}(\cdot,t)\,dt\right\|_{L^{2}(\Omega)}^{2}=0.$
(4.9)

We also notice that, in the weak sense (see e.g. [MR2597943, page 285]),

$\displaystyle\int_{0}^{2T_{\star}}\phi^{\prime}_{\eta}(t)\,u_{1}(\cdot,t)\,dt=\int_{0}^{T_{\star}-\eta}\phi^{\prime}_{\eta}(t)\,u_{1}(\cdot,t)\,dt+\int_{T_{\star}+\eta}^{2T_{\star}}\phi^{\prime}_{\eta}(t)\,u_{1}(\cdot,t)\,dt$
$\displaystyle\qquad=\int_{0}^{T_{\star}-\eta}\phi^{\prime}_{\eta}(t)\,u_{0}(\cdot,t)\,dt+\int_{T_{\star}+\eta}^{2T_{\star}}\phi^{\prime}_{\eta}(t)\,\widetilde{u}_{1}(\cdot,t)\,dt$
$\displaystyle\qquad=-\int_{0}^{T_{\star}-\eta}\phi_{\eta}(t)\,\partial_{t}u_{0}(\cdot,t)\,dt-\int_{T_{\star}+\eta}^{2T_{\star}}\phi_{\eta}(t)\,\partial_{t}\widetilde{u}_{1}(\cdot,t)\,dt$
$\displaystyle\qquad=-\int_{0}^{2T_{\star}}\phi_{\eta}(t)\,v(\cdot,t)\,dt,$

where

$v(\cdot,t):=\begin{cases}\partial_{t}u_{0}(\cdot,t)&{\mbox{ if
}}t\in[0,T_{\star}],\\\ \partial_{t}\widetilde{u}_{1}(\cdot,t)&{\mbox{ if
}}t\in(T_{\star},2T_{\star}].\end{cases}$

We insert this information into (4.9), finding that

$\lim_{\eta\searrow
0}\left\|\int_{0}^{2T_{\star}}\phi^{\prime}(t)\,u_{1}(\cdot,t)\,dt+\int_{0}^{2T_{\star}}\phi_{\eta}(t)\,v(\cdot,t)\,dt\right\|_{L^{2}(\Omega)}^{2}=0.$
(4.10)

Additionally,

$\displaystyle\left|\int_{0}^{2T_{\star}}\phi_{\eta}(t)\,v(\cdot,t)\,dt-\int_{0}^{2T_{\star}}\phi(t)\,v(\cdot,t)\,dt\right|\leqslant\|\phi\|_{L^{\infty}(\mathds{R})}\int_{T_{\star}-2\eta}^{T_{\star}+2\eta}|v(\cdot,t)|\,dt$
$\displaystyle\qquad\leqslant\|\phi\|_{L^{\infty}(\mathds{R})}\sqrt{4\eta\int_{T_{\star}-2\eta}^{T_{\star}+2\eta}|v(\cdot,t)|^{2}\,dt}$

and consequently

$\displaystyle\lim_{\eta\searrow
0}\left\|\int_{0}^{2T_{\star}}\phi_{\eta}(t)\,v(\cdot,t)\,dt-\int_{0}^{2T_{\star}}\phi(t)\,v(\cdot,t)\,dt\right\|_{L^{2}(\Omega)}^{2}$
$\displaystyle\qquad\leqslant 4\lim_{\eta\searrow
0}\eta\|\phi\|_{L^{\infty}(\mathds{R})}^{2}\iint_{\Omega\times(T_{\star}-2\eta,T_{\star}+2\eta)}|v(x,t)|^{2}\,dx\,dt$
$\displaystyle\qquad\leqslant 4\lim_{\eta\searrow
0}\eta\|\phi\|_{L^{\infty}(\mathds{R})}^{2}\Big{(}\|\partial_{t}u_{0}\|^{2}_{L^{2}((T_{\star}-2\eta,T_{\star}),\,L^{2}(\Omega))}+\|\partial_{t}\widetilde{u}_{1}\|^{2}_{L^{2}((T_{\star},T_{\star}+2\eta),\,L^{2}(\Omega))}\Big{)}$
$\displaystyle\qquad=0,$

thanks to (4.1).

This and (4.10) give that

$\left\|\int_{0}^{2T_{\star}}\phi^{\prime}(t)\,u_{1}(\cdot,t)\,dt+\int_{0}^{2T_{\star}}\phi(t)\,v(\cdot,t)\,dt\right\|_{L^{2}(\Omega)}^{2}=0,$

whence

$\int_{0}^{2T_{\star}}\phi^{\prime}(t)\,u_{1}(\cdot,t)\,dt=-\int_{0}^{2T_{\star}}\phi(t)\,v(\cdot,t)\,dt.$

This gives that $v=\partial_{t}u_{1}$ in the weak sense (see e.g. [MR2597943,
page 285]).

Now we adopt the short notation in (2.13) and use that $u_{0}$ is a solution
in $[0,T_{\star}]$ and $\widetilde{u}_{1}$ is a solution in
$[T_{\star},2T_{\star}]$: thus (see [MR2597943, page 352]), we have that, for
a.e. $t\in[0,2T_{\star}]$ and all $\psi\in H^{1}_{0}(\Omega)$,

$\displaystyle\langle\partial_{t}u_{1}(\cdot,t),\psi\rangle_{H^{-1}(\Omega),\,H^{1}_{0}(\Omega)}+\int_{\Omega}\nabla
u_{1}(x,t)\cdot\nabla\psi(x)\,dx-\int_{\Omega}f_{u_{1}}(x,t)\,\psi(x)\,dx$
$\displaystyle\qquad=\langle
v(\cdot,t),\psi\rangle_{H^{-1}(\Omega),\,H^{1}_{0}(\Omega)}+\int_{\Omega}\nabla
u_{1}(x,t)\cdot\nabla\psi(x)\,dx-\int_{\Omega}f_{u_{1}}(x,t)\,\psi(x)\,dx$
$\displaystyle\qquad=\begin{cases}&\displaystyle\langle\partial_{t}u_{0}(\cdot,t),\psi\rangle_{H^{-1}(\Omega),\,H^{1}_{0}(\Omega)}+\int_{\Omega}\nabla
u_{0}(x,t)\cdot\nabla\psi(x)\,dx-\int_{\Omega}f_{u_{0}}(x)\,\psi(x)\,dx\\\
&\qquad{\mbox{ for a.e. }}t\in[0,T_{\star}],\\\
&\displaystyle\langle\partial_{t}\widetilde{u}_{1}(\cdot,t),\psi\rangle_{H^{-1}(\Omega),\,H^{1}_{0}(\Omega)}+\int_{\Omega}\nabla\widetilde{u}_{1}(x,t)\cdot\nabla\psi(x)\,dx-\int_{\Omega}f_{\widetilde{u}_{1}}(x)\,\psi(x)\,dx\\\
&\qquad{\mbox{ for a.e. }}t\in[T_{\star},2T_{\star}],\\\ \end{cases}$
$\displaystyle\qquad=0,$

yielding the desired result in (4.8).

Recursively one also proves that $u_{k}$ is a solution of (4.6) in the time
interval $[0,(k+1)T_{\star}]$ for all $k\geqslant 2$, thus completing the
proof of Lemma 4.2. ∎

###  4.2. The iteration argument

We can now complete the proof of the global in time existence result:

###### Proof of Theorem 1.3.

Under the assumptions of Theorem 1.3, one obtains from Theorem 1.1 a solution
for a time interval $[0,T_{\star}]$, with $T_{\star}>0$ depending only on $n$,
$\Omega$, $\|K\|_{L^{2}(\Omega\times\Omega)}$,
$\|\omega\|_{L^{\infty}(\Omega\times(0,1))}$, and
$\|\beta\|_{W^{1,\infty}(\mathds{R})}$.

Thus, one defines recursively $u_{k}$ as in (4.7) and, in light pof Lemma 4.2,
obtains a solution in the time interval $[0,(k+1)T_{\star}]$ for all
$k\in\mathds{N}$ with $k\geqslant 1$, as desired. ∎

## References