\documentclass[reqno]{amsart}
\usepackage{epic, hyperref}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 119, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2005 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2005/119\hfil Nonautonomous ultraparabolic equations]
{Nonautonomous ultraparabolic equations applied to population
dynamics}
\author[N. Ghouali, T. M. Touaoula \hfil EJDE-2005/??\hfilneg]
{Noureddine Ghouali, Tarik Mohamed Touaoula}  % in alphabetical order

\address{Noureddine Ghouali  \hfill\break
Department of Mathematics, University of Tlemcen \\
BP 119, Tlemcen 13000, Algeria}
\email{n\_ghouali@mail.univ-tlemcen.dz}

\address{Tarik Mohamed Touaoula \hfill\break
Department of Mathematics, University of Tlemcen \\
BP 119, Tlemcen 13000, Algeria} 
\email{touaoulatarik@yahoo.fr}

\date{}
\thanks{Submitted August 15, 2005. Published October 26, 2005.}
\subjclass[2000]{35G10, 35K70}
\keywords{Ultraparabolic equations; perturbation; dynamics of the fish larvae}

\begin{abstract}
 We prove the existence and positivity of solutions to
 nonautonomus ultraparabolic equations using a
 perturbation method. These equations come from
 population dynamics, namely from a fish larvae model.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{lemma}[theorem]{Lemma}


\section{Introduction}\label{sec:tlemcen}

In the present work, we investigate a model for the dynamics of
the fish larvae of certain species, namely the equation
\begin{equation}
\frac{\partial w}{\partial t}+\mathop{\rm div}(Vw)
-\frac{\partial}{\partial x_3} \big( h\frac{\partial w}{\partial
x_3} \big) +\mu w=0,
\end{equation}
where $h,V,\mu$ depend on the time and space variables.

 The main characteristic of this equation is that it has mixed
parabolic-hyperbolic type, due to the directional separation of
the diffusion and convection effects. Such problem is called
also nonautonomous ultraparabolic equation, that is parabolic in
many directions.
 In a previous work by  Ghouali and Touaoula
\cite{Touaoula}, a simplified version of the model of the larvae
had been investigated. It was assumed that the horizontal current
$V_i$, $i=1,2$ is uniform throughout the water column, i.e. does not
depends of the vertical variable $x_3$. Under this assumption, it
was possible to uncouple the vertical and the horizontal
components in the following sense: The study was restricted to
each of the horizontal streamlines. Such restriction to a line
reduces the functions of time horizontal components to functions
of time alone so that the full model reduces on such a line to a
diffusion equation in the vertical variable coupled with a first
order growth equation. Our purpose in this work is to extend this
method to the more realistic situation where the horizontal
current depends of all its variables.

 The lack of the coercivity of the operator can be
handled by using a convenient perturbation argument. Monotone
operator theory \cite[p. 316]{LIONS1} can be applied  which gives
us existence and uniqueness of the solution of the perturbed
problem. After that,  we establish the positivity of our
solution. Then passing to the limit, in a suitable way, we obtain the
existence of a solution of the main model.

 The paper is organized as follows: Section 2 is devoted
to recall some important results. In section 3 we formulate the
perturbed problem. In section 4 we prove existence, uniqueness
and positivity of the solution. Section 5 is devoted to prove the
convergence result and the existence of the exact solution of the
model.

\section{Notation and preliminary results}\label{sec:tlemcen1}

We recall here some definitions and results that will be used latter.
Let $X$ be a separable and reflexive Sobolev space
with norm $\|\cdot\|$ and its dual $X'$ with norm $\|\cdot\|_*$. We denote
by $\langle ,\rangle$ the duality bracket of $X'\times X$.
For $v \in L^2(0,T;X)$, we define the norm
$$
\Big(\int^T_0\|v\|^2dt\Big)^{1/2}.
$$
We denote by $\mathcal{D}(0,T;X)$ the space
of infinitely differentiable functions with compact support in
$(0,T)$ and with values in $X$. We denote by $\mathcal{D'}(0,T;X)$ the
space of distributions on $(0,T)$ with values in $X$. We set also
$ W(0,T;X,X'):=\{v, v\in L^2(0,T;X),\frac{\partial v}{\partial
t}\in L^2(0,T;X')\}$.

\begin{definition} \rm
We say that an operator $A$ from $X$ to $X'$ is monotone, if
\begin{equation}\label{pos}
\langle A(u)-A(v),u-v \rangle  \geq 0\quad  \forall u,v \in X.
\end{equation}
The operator $A$ is strictly monotone if we have a strict
positivity in \eqref{pos} for all  $u,v \in X$ and $u\neq v$.
\end{definition}

\begin{remark} \rm
If $A$ is a linear operator, then the monotonicity is equivalent
to
$$
\langle Au,u \rangle \geq 0 \quad  \forall  u \in D(A).
$$
\end{remark}

\begin{definition} \rm
Let $A$ be a monotone operator from $X$ to $X'$. We say that $A$
is a maximal monotone operator if its graph is a maximal subset of
$ X\times X' $ with respect to set inclusion.
\end{definition}

\begin{lemma}[\cite{LIONS1}]
Let $L$ be a unbounded linear operator, with a dense domain $D(L)$
in $X$ taking its values in $X'$. Then $L$ is maximal monotone if
and only if $L$ is a closed operator and such that
\begin{gather*}
\langle Lv,v \rangle \geq 0 \quad \forall  v \in D(L),\\
\langle L^*v,v\rangle \geq 0 \quad  \forall  v \in D(L^*).
\end{gather*}
where $L^*$ is the adjoint operator of $L$.
\end{lemma}

\begin{theorem}[\cite{LIONS1}] \label{th:1}
Let $X$ be a reflexive Banach space. Let $L$ be a linear operator
of dense domain $D(L)\subset X$ and take its values in $X'$.
Assume that $L$ is maximal monotone and suppose that $A$ is a
monotone, coercive operator from $X$ to $X'$, i.e.
$$
\frac{\langle A(v),v\rangle }{\|v\|}\to\infty  \quad\mbox{as }
\|v\|\to\infty.
$$
Then, for all $f \in X'$, there exists
$u \in D(L)$ such that $Lu+A(u)=f$.
\end{theorem}

\begin{remark}\label{rm:2} \rm
If we assume in addition that the operator $A$ is strictly
monotone then there exists a unique solution $u \in D(L)$ such
that $Lu+A(u)=f$.
\end{remark}

\begin{remark} \rm
One can easily see that in the case of a linear operator $A$, the
coercivity implies strictly monotonicity.
\end{remark}

\begin{remark}\label{cor:harnack7} \rm
Let $u$ be a solution to the  problem
\begin{equation}\label{aaa}
\begin{gathered}
\frac{\partial u}{\partial t}+Au=f, \\
u(0)=u_0,
\end{gathered}
\end{equation}
where $A$ is a linear operator. We set $u=ve^{k t}$,
$k \in \mathbb{R}$, then $v$ is a solution of the problem
\begin{equation}
\begin{gathered}
v'(t)+\left( A+k I\right) v(t)=f_1,\\
v(0)=u_0.
\end{gathered}\label{11}
\end{equation}
Hence, proving existence, uniqueness and positivity of solutions
of problem (\ref{11}) is equivalent to prove the same properties
to problem (\ref{aaa}). Throughout this paper we will deal with
problem (\ref{11}), where $k$ is a real constant that we will
choose later.
\end{remark}

\begin{remark}\label{rm:21} \rm
 We consider two Hilbert spaces
$V$,$H$ with $V \hookrightarrow H$,  the continuous injection
$\hookrightarrow $ having dense image in H. Then we can identify
$H$ with its dual $H'$, and therefore
$$
V\hookrightarrow H\hookrightarrow V'.
$$
\end{remark}

 From Remarks \ref{cor:harnack7} and \ref{rm:21} we obtain
the following Lemma.

\begin{lemma}[\cite{DAUTRAY}]
For $u_0 \in H $ there exists $v$ in $W(0,T;V,V')$, such that
$v=u_0 $ in $H$. Thus $w=u-v$, solves the  problem
\begin{equation}\label{aa}
\begin{gathered}
\frac{\partial w}{\partial t}+(A+kI)w=f_2, \\ w(0)=0,
\end{gathered}
\end{equation}
where $u$ is solution of problem \eqref{11}.
\end{lemma}
Therefore, we will consider the case where $u_0\equiv 0$.

\section{The model}

Our model takes into account both the physical and biological
effects. For the physical part, the model  stresses
two main factors: 1) Transport entailed by the currents: The
currents are computed using Navier-Stokes equations and are
introduced in the equations of the larvae as functions of space
and time with sufficient regularity to allow existence and
uniqueness of stream lines.
2) Vertical diffusion induced by vertical mixing in the upper part of
the water column. For the biological part the main parameters are
a function which gives the instantaneous rate of progression
within the stages from the egg fertilization to the end of the
yolk-sac period.

The model is expressed in a generality which encompasses a large
variety of situations. The motivation for this work
is the study of the dynamics on the Bay of Biscay anchovy; that is,
 a region of the Atlantic ocean close
to the French coast, bordered eastward by the continental shelf.
The Bay of Biscay goes from the Northern Spanish coast up to about
$46^\circ$ in ``latitude''. In this region at the end of May, a
thermocline establishes. The top of the thermocline is
roughly at the same distance $z_{\rm therm}$ from the surface. The
thermocline divides the water column into three regions:
The upper part, from the surface to $z_{\rm therm}$ meters deep,
is the so called mixed layer and its where the larvae grow.
Below this there is the thermocline, a rather thin layer where
the temperature loses rapidly a few degrees, and the vertical
mixing coefficient is negligibly small. Below the thermocline is
another well mixed layer where the temperature changes very little
with depth. This region is of no concern to us for this
study. We will be confined to the mathematical
issues related to the above model, and we study only the upper
layer, the mixed layer of the water column; see Figure \ref{fig1}.

\begin{figure}[t]
\begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(50,35)(0,0)
\put(16,28){water surface}
\drawline(0,26)(50,26)
\put(0,17){$z_{\rm therm}\Bigg\{$}
\put(17,19){mixed layer,}
\put(17,15){where larvae grow}
\dashline{1}(0,9)(50,9)
\put(16,5){thermocline}
\dashline{1}(0,2.5)(50,2.5)
\put(10,-2){another mixed layer}
\end{picture}
\end{center}
\caption{Water column divided into three regions \label{fig1}}
\end{figure}

The domain under consideration is $\Omega =D\times (0,z^{\ast })$,
where $D $ is an open subset of the surface, that is $D$ is a
portion of the plane, and $z^{\ast }$ is the distance from the
surface to a region above the thermocline.

 We denote by $Q$ the product space $ \Omega\times (0,T)$ and
$ \Sigma:=\Gamma\times (0,T)$ the boundary of $Q$.
 The state variable for the dynamics of the larvae is
the density of larvae. For the part of the larval cycle which goes
from fertilization to the end of stage, the density
$w=w(t,s,P)$, where $s$ denotes the position within the stages,
which we take specifically of the Bay of Biscay anchovy in
$[1,12[$ see for example \cite{Arino} and $P=(x_1,x_2,x_3)$
represents a generic point in the physical space.

 The region of observation is assimilated to the product of the
horizontal plane and a vertical line. The origin is a point of the
surface in the sea, the $x_1$ axis is oriented westward, the $x_2$
axis is oriented northward, and the $x_3$ axis is oriented
downward. Of course $t$ is the chronological time. $w$ is a
density with respect to the stage and the position. The larvae are
characterized by their density, that is to say, at each time $t\in
[ 0,T] $, where $T$ is the maximal time of observation, $w(t,s,P)$
can be thought of as the larvae biomass per unit of volume
evaluated at the point $P$, at that time. The full model is as
follows
\begin{equation}
\begin{gathered}
\frac{\partial w}{\partial t}+\frac{\partial ( fw) }{\partial s}
+div\,(Vw)-\frac{\partial }{\partial x_3}( h\frac{\partial
w}{\partial x_3}) +\gamma w=0,  \\
w=0, \quad \mbox{in } \Sigma \\
 w(t,1,x,y,z)=l_0(t,P).
\end{gathered} \label{100}
\end{equation}
The significance of the parameters in the model is as
follows:

\noindent The \textbf{velocity} vector
$V(t,P)=(V_{1}(t,P),V_{2}(t,P),V_{3}(t,P)) $ describes the sea
current which is supposed to be known.

\noindent The \textbf{mixing coefficient} $h=h(t,P)$ gives the
diffusion rate, supposed to be essentially vertical.

\noindent The  \textbf{growth function} is the main biological
parameter, $f(t,s)$, which gives the instantaneous
rate of progression within the stages from the egg fertilization
to the end of the yolk-sac period. For the principle of
determination see \cite{Motos,Arino}.

\noindent The \textbf{mortality of larvae} is
modelled by the expression $\gamma =\gamma(t,s,P)$.

\noindent The \textbf{Demographic boundary conditions}
 are given at $s=1$, at any time during the
spawning period, the variable $s$ takes its values in the interval
$[1,12[$, where $s=1$ corresponds to the newly fertilized eggs,
and $s=12$, to the end of the yolk sac period.

\noindent The \textbf{boundary condition} is zero, we assume
that there is no larvae on the boundary.

\noindent The \textbf{Time of observation} is restricted to
 time interval when the larvae remains in the domain $\Omega$.

\section{Existence, uniqueness and positivity of solution of
the perturbed problem}

The objective of this section is to study existence, uniqueness
and positivity of solution of the associated perturbed problem
\eqref{111}. For this, we start by using the method of
characteristics to reduce the number of variables. We assume that
\begin{itemize}
\item[(H1)] $f$ is in  $C^{1}((0,T)\times (1,12))$.
\end{itemize}
We introduce the flow generated by the size growth, that is
\[
\phi:=\phi (\tau ,t_{0},1),
\]
and for each initial value
 $\tilde{\zeta}\equiv(t_{0},1)$, $\phi (\tau
,\tilde{\zeta})$ is the solution of the equation
\begin{equation}
\big( \frac{dt}{d\tau },\frac{ds}{d\tau }\big) =(1,\;f(t,s)),
\label{caracte}
\end{equation}
that satisfies $t(0)=t_{0}$, $s(0)=1$, since the theory of ordinary
differential equations guarantees that a unique characteristic
curve passes through each point $\tilde{\zeta}$. Let
\[
t=T(\tau ,t_{0}), \quad s=S(\tau ,t_{0}),
\]
be a solution of the characteristic system (\ref{caracte})
emanating from the point $\tilde{\zeta}$. We assume that
$$
\frac{\partial S}{\partial t_0}-f\neq 0
$$
at $\tau=0$.
Without loss of generality we can assume that $t_0=0$, otherwise
we replace $V$ and $h$ by those restrictions along the
characteristic line.
 So to each $\tilde{\zeta}$, we have associated the
following problem, see for instance \cite{Touaoula},
\begin{equation}
\begin{gathered}
\frac{\partial l}{\partial t}+\mathop{\rm div}(Vl)
-\frac{\partial }{\partial z}\big( h\frac{\partial l}{\partial z}\big)
+(\mu+k)l=0, \\
l=0, \quad \mbox{in } \Sigma, \\
l(0,P)=l_0(P),
\end{gathered} \label{110}
\end{equation}
where $\mu$ is function of order zero, and $k$ is a real constant
that we will chose later. We will use a perturbation method to get
a time dependent parabolic equation whose resolution will yield to
the solution of equation (\ref{100}). Namely we consider for the
perturbed problem
\begin{equation}\label{111}
\begin{gathered}
\frac{\partial l}{\partial t}+\mathop{\rm div}(Vl)
-\sum_{i=1}^{3}\frac{\partial }{\partial x_{i} }
\big( a_{i}^{\varepsilon}\frac{\partial l}{\partial x_{i}}
\big) +(\mu+k) l=0, \\
l=0,\quad \mbox{in } \Sigma, \\
 l(0,P)=l_0(P),
\end{gathered}
\end{equation}
where
$$
a_{i}^{\varepsilon}(t,P)=\begin{cases}
\varepsilon   &\mbox{if    } i=1,2 \\
h(t,P)+\varepsilon &\mbox{if    } i=3.
\end{cases}
$$
Let
$$
Lu_{\varepsilon}=\frac{\partial u_{\varepsilon}}{\partial t}
+\mathop{\rm div}(Vu_{\varepsilon})+(k+\mu)u_{\varepsilon},
$$
with
 $$
D(L)=\{ v\in L^{2}(0,T;W_0^{1,2}(\Omega));\frac{\partial v}{\partial
t}\in L^{2}(0,T;W^{-1,2}(\Omega)), v(0)=0\},
$$
and
$$
Au_{\varepsilon}=-\sum_{i=1}^{3}\frac{\partial }{\partial x_{i} }
\big( a_{i}^{\varepsilon}\frac{\partial u_{\varepsilon}}{\partial
x_{i}}\big),
$$
defined by
$$
\langle Au_{\varepsilon},v\rangle
=\sum_{i=1}^{3}\int_{Q}
a_{i}^{\varepsilon}\frac{\partial u_{\varepsilon}}{\partial
x_{i}}\frac{\partial v}{\partial x_{i}}\,dP\,dt,
$$
for each $v \in L^2(0,T;W_0^{1,2}(\Omega))$. We now state the assumptions
of this
section.
\begin{itemize}
\item[(H2)] $h \in C^1(\bar{Q}))$, ${V}_{i}\in C([0,T]\times
\bar{\Omega})$, $i=1,2,3$ and ${\gamma}\in
C([0,T]\times[1,12]\times \bar{\Omega)}$.

\item[(H3)] $h\geq c_0>0 $ in $[0,T]\times \bar{\Omega}$.

\end{itemize}
The main result of this section is the following theorem that
gives conditions under which problem \eqref{111} has a unique
positive solution.

\begin{theorem}
Assume (H2)--(H3) hold. Let $l_0\in L^{2}(\Omega)$, be such that $l_0\geq 0$.
 Then problem \eqref{111} has a unique non negative solution
$u_{\varepsilon}
\in D(L)$.
\end{theorem}

\begin{proof}
The main idea is to use Theorem \ref{th:1}.
In the first step we will see that $L$ is a closed operator with a
dense domain; indeed, let $u_n$ in $D(L)$ be such that
$u_n\to u$ in $L^2(0,T;W_0^{1,2}(\Omega))$ and
$Lu_n\to y$ in $L^2(0,T;W^{-1,2}(\Omega))$, hence
$$
u_n\to u \quad \mbox{in } \mathcal{D'}(0,T;W^{-1,2}(\Omega))
$$
and
$$
Lu_n\to y \quad \mbox{in }
\mathcal{D'}(0,T;W^{-1,2}(\Omega)).
$$
It follows that
$$
Lu_n\to Lu \quad \mbox{in   } \mathcal{D'}(0,T;W^{-1,2}(\Omega)).
$$
Therefore,  $y=Lu$ and $u \in D(L)$. Hence $L$ is a closed operator. It is
not difficult to see that $\mathcal{D}(0,T;W_0^{1,2}(\Omega))$ is
included in $D(L)$, then  we deduce that $D(L)$ is dense in
$L^2(0,T;W_0^{1,2}(\Omega))$. Concerning the monotonicity of $L$,
we have for $u \in D(L)$,
\begin{align*}
&\langle Lu,u \rangle \\
&=\int_{0}^{T}\langle \frac{\partial u}{\partial t},u\rangle dt
+\int_{Q}(\mathop{\rm div}(Vu)u+(k+\mu)u^{2})\,dP\,dt,\\
&=\frac{1}{2}\int_{0}^{T}\frac{d}{dt}\|u(t)\|_{*}^{2}dt
+\int_{\Sigma}(V,\eta)u^{2}d\sigma-\int_{Q}(V,\nabla u)u\,dP\,dt
+\int_{Q}(k+\mu)u^{2}\,dP\,dt,\\
&=\frac{1}{2}\|u(T)\|_{*}^{2}-\frac{1}{2}\int_{Q}(V,\nabla
u^{2})\,dP\,dt+\int_{Q}(k+\mu)u^{2}\,dP\,dt,
\end{align*}
with $\eta$ is exterior normal, and $(,)$ is the scalar product.
Hence by integration by parts we obtain that
\[
\langle Lu,u\rangle =\frac{1}{2}\|u(T)\|_{*}^{2}
+\int_{Q}(k+\frac{1}{2}\mathop{\rm div}(V)+\mu)u^{2}\,dP\,dt,\\
\]
choosing $k$ large so that
$$
k+\frac{1}{2}\mathop{\rm div}(V)+\mu\geq 0,
$$
it follows that $L$ is monotone for all $ u \in D(L)$. In addition
for $u \in D(L)$,
\begin{align*}
\langle Lu,v \rangle
&=\int_{0}^{T}\langle \frac{\partial u}{\partial t},v\rangle dt
+\int_{Q}(\mathop{\rm div}(Vu)v+(k+\mu)uv)\,dP\,dt,\\
&= \int_{0}^{T}\langle u,-\frac{\partial v}{\partial t}\rangle dt
+\langle u(T),v(T)\rangle +\int_{Q}(-(V,\nabla v)+(k+\mu)v)u\,dP\,dt,
\end{align*}
thus, the associated adjoint operator is
$$
L^*v=-\frac{\partial v}{\partial t}-(V,\nabla v)+(k+\mu)v,
$$
with
$$
D(L^*)=\{ v\in L^{2}(0,T;W_0^{1,2}(\Omega));\frac{\partial
v}{\partial t}\in L^{2}(0,T;W^{-1,2}(\Omega)), v(T)=0\}.
$$
The proof of monotonicity of $L^*$ is similar to the one of $L$.
Then $L$ is a maximal monotone operator.
 It remain to see that $A$ is coercive, indeed for
$u \in L^{2}(0,T;W_0^{1,2}(\Omega))$ and applying the hypothesis on $h$,
it holds
$$
\langle Au,u\rangle =\sum_{i=1}^{3}\int_{Q}
a_{i}^{\varepsilon}|\frac{\partial u}{\partial x_{i}}|^{2}\,dP\,dt
\geq M_{\varepsilon}\|u\|_{L^{2}(0,T;W_0^{1,2}(\Omega))}^{2}.
$$
According to Theorem \ref{th:1} and Remark \ref{rm:2}, we get the
existence of a unique solution $u_{\varepsilon}\in D(L)$ of the
perturbed problem \eqref{111}. Hence for all
$v \in L^{2}(0,T;W_0^{1,2}(\Omega))$, we have
\begin{equation}\label{perturb}
\int_{0}^{T}\langle \frac{\partial u_{\varepsilon}}{\partial
t},v\rangle dt+\int_{Q}(\mathop{\rm div}(Vu_{\varepsilon})+(\mu
+k)u_{\varepsilon})v\,dP\,dt+\sum_{i=1}^{3}\int_{Q}
a_{i}^{\varepsilon}\frac{\partial u_{\varepsilon}}{\partial
x_{i}}\frac{\partial v}{\partial x_{i}}\,dP\,dt=0.
 \end{equation}
We prove now the positivity of the solution. We set
$u=u^{+}-u^{-}$, where $u^{+}$ and $u^{-}$ are
respectively the positive and negative part of $u$.
Using $u^-_{\varepsilon}$ as a test function in
\eqref{perturb} and integrating on $(0,t)$, we get
\begin{align*}
&-\int_{0}^{t}\langle \frac{\partial u^-_{\varepsilon}}{\partial
t},u^-_{\varepsilon}\rangle dt-\int^t_0\int_{\Omega}
(\mathop{\rm div}(Vu^-_{\varepsilon})+(\mu
+k)u^-_{\varepsilon})u^-_{\varepsilon}\,dP\,dt \\
&-\sum_{i=1}^{3}\int^t_0\int_{\Omega}
a_{i}^{\varepsilon}|\frac{\partial u^-_{\varepsilon}}{\partial
x_{i}}|^2\,dP\,dt=0,
\end{align*}
integrating by parts two times, we obtain
\[
-\frac{1}{2}\|u_{\varepsilon}^-(t)\|_{*}^{2}
 =  \int^t_0\int_{ {\Omega}}(\frac{1}{2}\mathop{\rm
div}(V)+\mu+k)(u_{\varepsilon}^-)^{2}\,dP\,dt+
\sum_{i=1}^{3}\int^t_0\int_{\Omega}
a_{i}^{\varepsilon}|\frac{\partial u^-_{\varepsilon}}{\partial
x_{i}}|^2\,dP\,dt;
\]
hence,
$-\frac{1}{2}\|u_{\varepsilon}^-(t)\|_{*}^{2}\geq 0$.
 Then $u_{\varepsilon}^-(t)=0$ for all $t\in(0,T)$. The proof is
complete.
\end{proof}

\section{The exact solution}

In this section we show that the perturbed solution defined in
\eqref{perturb}
 tends to the desired solution of problem (\ref{110}) in
$L^{2}(Q)$ as $\varepsilon$ tends to $0$. Our main result is the
following Theorem.

\begin{theorem}
Let $l_0 \in L^{2}(\Omega)$ and consider $u_{\varepsilon}$ the
solution to problem \eqref{111}, then $u_{\varepsilon}$  converges
weakly to $u$ in $L^2(Q)$ where $u$ is a distributional solution
of the problem \eqref{110}. In addition we have
$\frac{\partial u}{\partial x_3}\in L^2(Q)$ and $u$ satisfies
\begin{equation}\label{eq:sss}
\begin{aligned}
&-\int_{\Omega}\int_{0}^{T}u\frac{\partial \phi}{\partial t}dx
dt+\int_Q(-(V,\nabla
\phi)+(\mu+k)\phi)u\,dP\,dt+\int_{Q}h\frac{\partial u}{\partial
x_3}\frac{\partial \phi}{\partial
x_3}\,dP\,dt\\
&=\int_{\Omega}l_0(P)\phi(0,P)dP,
\end{aligned}
\end{equation}
for all $\phi \in K$ where
\begin{equation}\label{eq:kkk}
K\equiv \{\phi \in L^2(0,T;W_0^{1,2}(\Omega)):\frac{\partial
\phi}{\partial t}\in L^2(0,T;W^{-1,2}(\Omega))\cap L^2(Q),\;
\phi(T)=0\}.
\end{equation}
\end{theorem}

\begin{proof}
 By taking $u_{\varepsilon}$ as a test function in
\eqref{perturb}, we obtain
\begin{equation}\label{perturb1}
\int_{0}^{T}\langle \frac{\partial u_{\varepsilon}}{\partial
t},u_{\varepsilon}\rangle dt+\int_{Q}(\mathop{\rm
div}(Vu_{\varepsilon})+(\mu+k)u_{\varepsilon})u_{\varepsilon}\,dP\,dt
+\sum_{i=1}^{3}\int_{Q} a_{i}^{\varepsilon}
|\frac{\partial u_{\varepsilon}}{\partial x_{i}}|^2\,dP\,dt=0,
\end{equation}
integrating by parts and using the definition of
$u_{\varepsilon}$, we deduce
\[
\frac{1}{2}\|u_{\varepsilon}(T)\|_{*}^{2}+\int_{Q}(k+\frac{1}{2}\mathop{\rm
div}(V)+\mu)u_{\varepsilon}^{2}\,dP\,dt+\sum_{i=1}^{3}\int_{Q}
a_{i}^{\varepsilon}\big|\frac{\partial u_{\varepsilon}}{\partial
x_{i}}\big|^2\,dP\,dt=\frac{1}{2}\|l_0\|_{*}^{2}.
\]
Since $\mathop{\rm div}(V)$ and $\mu $ are bounded functions we
conclude that $\|u_{\varepsilon}\|_{L^{2}(Q)}^{2}\leq C$ and then
there exists a subsequence called also $u_{\varepsilon}$ such that $
u_{\varepsilon}\rightharpoonup u$ weakly in $L^{2}(Q)$. Notice
that, in the same way, we obtain that
\[
\sum_{i=1}^{3}\int_{Q} a_{i}^{\varepsilon}|\frac{\partial
u_{\varepsilon}}{\partial x_{i}}|^{2}\,dP\,dt\leq C_{1}.
\]
By letting $\varepsilon\to 0$, we obtain
\begin{equation}\label{d1dd}
\limsup_{\varepsilon\to 0}\int_{Q}h|\frac{\partial
u_{\varepsilon}}{\partial x_3}|^{2}\,dP\,dt\leq C_{1}.
\end{equation}
We claim that $u$ is a solution of \eqref{110} in the sense of
distribution. To proof the claim we consider $\phi\in
\mathcal{C}^{\infty}_0(\Omega\times (0,T))$, then using $\phi$ as
a test function in \eqref{111} we obtain
\begin{align*}
&-\int_{\Omega}\int_{0}^{T}u_{\varepsilon}\phi_t \,dP\,dt+\int_Q(-(V,\nabla
\phi)+(\mu+k)\phi)u_{\varepsilon} \,dP\,dt\\
&+\sum_{i=1}^{3}\int_{Q} u_{\varepsilon}\frac{\partial }{\partial x_{i} }
\big(a_{i}^{\varepsilon}\frac{\partial \phi}{\partial x_{i}}\big)
\,dP\,dt\\
&=\int_{\Omega}l_0(P)\phi(0,P)dP.
\end{align*}
Since $\nabla h\in (L^2(Q))^3$ and $u_{\varepsilon}\rightharpoonup u$
weakly in $L^2(Q)$, passing to the limit in the above equality we
obtain
\begin{align*}
&-\int_{\Omega}\int_{0}^{T}u\phi_t \,dP\,dt+\int_Q(-(V,\nabla
\phi)+(\mu+k)\phi)u \,dP\,dt+\int_{Q} u\frac{\partial }{\partial x_{3}
}\big( h\frac{\partial \phi}{\partial x_{3}}\big)
\,dP\,dt\\
&=\int_{\Omega}l_0(P)\phi(0,P)dP.
\end{align*}
Hence $u$ is a distributional solution to problem \eqref{110} and
the claim follows.

To get  more regularity on $u$ we set
\begin{eqnarray*}
\Psi_{\varepsilon}(t,x_3)=\int_{D}hu_{\varepsilon}\,dx\,dy,
\end{eqnarray*}
where $u_\varepsilon$ is the solution of \eqref{111}. Using the hypothesis
on $h$ and $V$ and by the classical result on the theory of
regularity we obtain that $u_{\varepsilon} \in C^1([0,T]\times
\bar{\Omega})$. Thus
\[
\frac{\partial \Psi_{\varepsilon}}{\partial
x_3}=\int_{D}(h\frac{\partial u_{\varepsilon}}{\partial
x_3}+u_{\varepsilon}\frac{\partial h}{\partial x_3})\,dx\,dy,
\]
by integrating over $(0,T)\times(0,z^{*})$ we get
\[
\int_{0}^{T}\int_{0}^{z^{*}}|\frac{\partial
\Psi_{\varepsilon}}{\partial x_3}|^{2}dx_3dt\leq
\int_{Q}h^{2}|\frac{\partial u_{\varepsilon}}{\partial
x_3}|^{2}\,dP\,dt+C\int_{Q}|u_{\varepsilon}|^{2}\,dP\,dt\leq C_{2}.
\]
Since $\Psi_{\varepsilon}$ is bounded in $L^{2}((0,T)\times
(0,z^*))$, which can be proved easily,  we conclude that $\Psi_\varepsilon$
is bounded in $L^2(0,T;W^{1,2}_0(0,z^*))$, hence up to a
subsequence, called also $\Psi_{\varepsilon}$, we obtain that
$\Psi_{\varepsilon}$ converges weakly in
$L^2(0,T;W^{1,2}_0(0,z^*))$ to $\Psi$ where
\begin{eqnarray*}
\Psi=\int_{D}h\, u\, \,dx\,dy.
\end{eqnarray*}
Note that the last identification follows by the fact that
$u_\varepsilon\rightharpoonup u$ in weak topology of $L^2(Q)$ and by the
uniqueness of the weak limit.

 We claim that $\frac{\partial u}{\partial x_3}\in L^2(Q)$. To show this
claim we prove that $\frac{\partial u}{\partial x_3}\in
(L^2(Q))'\equiv L^2(Q)$. Note that $\frac{\partial u}{\partial x_3}$ is well
defined as a distribution. Let $\phi\in
\mathcal{C}^{\infty}_0(Q)$, then we have
\begin{align*}
\int_{Q}\frac{\partial u}{\partial x_3}\phi \,dP\,dt
&=  -\int_{Q}\frac{\partial \phi}{\partial x_3}u \,dP\,dt\\
&= -\lim_{\varepsilon \to 0}\int_{Q}\frac{\partial \phi}{\partial x_3}u_{\varepsilon}
\,dP\,dt \\
&=\lim_{\varepsilon \to 0}\int_{Q}\frac{\partial
u_{\varepsilon}}{\partial x_3}\phi \,dP\,dt \\
& \le  \lim_{\varepsilon \to 0}\Big(\int_{Q}|\frac{\partial u_{\varepsilon}}{\partial
x_3}|^2\,dP\,dt\Big)^{1/2}\Big(\int_{Q}|\phi|^2\,dP\,dt\Big)
^{1/2}.
\end{align*}
Then we conclude that
$$
|\int_{Q}\frac{\partial u}{\partial x_3}\phi \,dP\,dt|\le
C\Big(\int_{Q}|\phi|^2\,dP\,dt\Big)^{1/2}
$$
for all $\phi\in \mathcal{C}^{\infty}_0(Q)$. Hence by density we
conclude that $\frac{\partial u}{\partial x_3}\in (L^2(Q))'\equiv L^2(Q)$ and
then the claim follows.

 Therefore we conclude that
\[
\int_{Q}h\frac{\partial u_{\varepsilon}}{\partial
x_3}\frac{\partial v}{\partial
x_3}\,dP\,dt\to\int_{Q}h\frac{\partial u}{\partial
x_3}\frac{\partial v}{\partial x_3}\,dP\,dt,
\]
for all $v \in L^{2}(0,T;W_0^{1,2}(0,z^{*}))$. Let $\phi \in
\mathcal{C}^{\infty}_0(Q)$, using a density result, see for
example \cite{SCH}, \cite{BOU}, we get that
$\{\eta(t,x_1,x_2)\times \psi(t,x_3)\}$ is a total family in
$\mathcal{C}^{\infty}_0(Q)$. Then by the above computation we
obtain that
\[
\int_{Q}h\eta\frac{\partial u_{\varepsilon}}{\partial
x_3}\frac{\partial \psi}{\partial x_3}\,dP\,dt\to\int_{Q}h\eta
\frac{\partial u}{\partial x_3}\frac{\partial \phi}{\partial
x_3}\,dP\,dt.
\]
Hence by the density result obtained in \cite{SCH} and in
\cite{BOU} we get the same conclusion for all $\phi \in
\mathcal{C}^{\infty}_0(Q)$; hence
\[
\int_{Q}h\frac{\partial u_{\varepsilon}}{\partial
x_3}\frac{\partial \phi}{\partial
x_3}\,dP\,dt\to\int_{Q}h\frac{\partial u}{\partial
x_3}\frac{\partial \phi}{\partial x_3}\,dP\,dt.
\]
Since $\phi \in \mathcal{C}^{\infty}_0(Q)$ is dense in
$L^{2}(0,T;W_0^{1,2}(D\times (0,z^{*})))$ and by the fact that
$\frac{\partial u}{\partial x_3}\in L^2(Q)$ we get that
\eqref{eq:sss} holds for all $\phi \in L^{2}(0,T;W_0^{1,2}(D\times
(0,z^{*})))$. By letting $\varepsilon$ tend to $0$ in
\eqref{perturb}, we obtain
\begin{align*}
& \int_Q u\frac{\partial \phi}{\partial t}dx dt+\int_Q(-(V,\nabla
\phi)+(\mu+k)\phi)u\,dP\,dt+\int_{Q}h\frac{\partial u}{\partial
x_3}\frac{\partial \phi}{\partial
x_3}\,dP\,dt\\
&=\int_{\Omega}l_0(P)\phi(0,P)dP,
\end{align*}
for all $\phi \in K$, where $K$ is defined in \eqref{eq:kkk}.
The proof is complete.
\end{proof}

Now, we give a remark on the uniqueness of solution. In
\cite{EVZ}, the authors consider the  Cauchy problem
\begin{equation}\label{fff}
u_t-\Delta_x u=\partial_y(f(u)), \:\:(x,y)\in \mathbb{R}^N,t>0\,.
\end{equation}
Using the vanishing viscosity argument and the notion of Entropy
solution, they obtain existence and uniqueness of solution to
problem \eqref{fff}. The argument used depends on the presence of
the linear operator $\Delta_x$ and the estimate obtained in
\cite{GR}.

 The extension of the above uniqueness result to a non-autonomous
problem seems to be a more difficult technical problem
and it will be treated in forthcoming work.

\subsection*{Acknowledgments}
The authors wish to thank the anonymous referee for his/her
helpful suggestions and comments.
The authors wish to dedicate this work to the memory of the late
professor Ovide Arino.

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\end{thebibliography}


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