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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 195, pp. 1--5.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}
\begin{document}
\title[\hfilneg EJDE-2017/195\hfil Remarks on stability]
{Remarks on the preceding paper by Crespo, Ivorra and Ramos on the
stability of \\ bioreactor processes}
\author[J. I. D\'iaz \hfil EJDE-2017/195\hfilneg]
{Jes\'us Ildefonso D\'iaz }
\address{Jes\'us Ildefonso D\'iaz \newline
Departamento de Matem\'atica Aplicada,
$\&$ Instituto de Matem\'atica Interisciplinar,
Universidad Complutense de Madrid,
Plaza de Ciencias, 3, 28040 Madrid, Spain}
\email{jidiaz@ucm.es}
\thanks{Submitted July 2, 2017. Published August 7, 2017.}
\subjclass[2010]{35K51, 47D03}
\keywords{ Asymptotic stability; advection-diffusion-reaction system;
\hfill\break\indent continuous bioreactor; accretive operators; smoothing effects for parabolic
equations}
\begin{abstract}
In this short note we indicate some improvement of an article published
in this journal by Crespo, Ivorra and Ramos \cite{CIR}. The techniques
used are connected with several smoothing effects associated with linear
partial differential operators which give rise to some accretive operators
in $L^1(\Omega )$, as well as with some $H^2(\Omega )$ estimates
independent on time.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\allowdisplaybreaks
\section{Introduction and results}
In the previous article in this journal Crespo, Ivorra and Ramos \cite{CIR}
obtained an interesting theorem on the stability of bioreactor processes
under suitable conditions. The main goal of this short note is to present
some improvement of their main result by using quite different techniques of
proof.
Let us consider the non-dimensional form of the system considered in
\cite{CIR},
\begin{equation}
\begin{gathered}
\frac{\partial S}{\partial t}=\frac{\sigma ^2}{{\rm Th}_S}
\frac{1}{r}\frac{\partial }{\partial r}(r\frac{\partial S}{\partial r})+
\frac{1}{{\rm Th}_S}\frac{\partial ^2S}{\partial z^2}+
\frac{1}{\rm Da}\frac{\partial S}{\partial z}-\mu (S)B \quad \text{in }
\Omega \times (0,T), \\
\frac{\partial B}{\partial t}=\frac{\sigma ^2}{{\rm Th}_B}
\frac{1}{r}\frac{\partial }{\partial r}(r\frac{\partial B}{\partial r})+
\frac{1}{{\rm Th}_B}\frac{\partial ^2B}{\partial z^2}+
\frac{1}{\rm Da}\frac{\partial B}{\partial z}+\mu (S)B \quad \text{in }
\Omega \times (0,T), \\
\frac{1}{{\rm Th}_S}\frac{\partial S}{\partial z}+\frac{1}{\rm
Da}S=\frac{1}{\rm Da} \quad \text{in }\Gamma _{\mathrm{in}
}\times (0,T), \\
\frac{1}{{\rm Th}_B}\frac{\partial B}{\partial z}+\frac{1}{\rm
Da}B=0 \quad \text{in }\Gamma _{\mathrm{in}}\times (0,T), \\
\frac{\partial S}{\partial r}=\frac{\partial B}{\partial r}=0 \quad \text{in }
\Gamma _{\mathrm{sym}}\times (0,T), \\
\frac{\partial S}{\partial r}=\frac{\partial B}{\partial r}=0 \quad \text{in }
\Gamma _{\mathrm{wall}}\times (0,T), \\
\frac{\partial S}{\partial z}=\frac{\partial B}{\partial z}=0 \quad \text{in }
\Gamma _{\mathrm{out}}\times (0,T),
\end{gathered} \label{eq:2_4}
\end{equation}
jointly with by the initial conditions
\begin{equation}
S(\cdot ,\cdot ,0)=S_{\mathrm{init}}\quad \text{and}\quad
B(\cdot ,\cdot ,0)=B_{\mathrm{init}}\quad \text{in }\Omega ,
\label{eq:2_5}
\end{equation}
where $\Omega =(0,1)\times (0,1)$ is the nondimensional domain,
$\Gamma _{\mathrm{in}}=(0,1)\times \{1\}$,
$\Gamma _{\mathrm{out}}=(0,1)\times \{0\}$,
$\Gamma _{\mathrm{wall}}=\{1\}\times (0,1)$ and
$\Gamma _{\mathrm{sym}}=\{0\}\times (0,1)$ are the non-dimensional boundary
edges. This system can
be more conceptually written as a special case of the system
\begin{equation}
\begin{gathered}
u_{t}-L_{1}u-f(u,v)=0 \quad \text{in }\Omega \times (0,T), \\
v_{t}-L_{2}v+f(u,v)=0 \quad \text{in }\Omega \times (0,T),\\
\frac{\partial u}{\partial n}+b_{1}(x)u=g(x) \quad
\text{on }\partial \Omega \times (0,T), \\
\frac{\partial v}{\partial n}+b_{2}(x)v=0
\quad \text{on }\partial \Omega \times (0,T),\\
u(x,0)=u_{0}(x) \quad \text{on }\Omega\\
v(x,0)=v_{0}(x) \quad \text{on }\Omega,
\end{gathered} \label{system}
\end{equation}
with obvious choices of the data. Which is relevant in our approach is that
$\Omega $ is a convex bounded open set of $R^{3}$ with piece-wise smooth
boundary and the elliptic operators are coercive and can be expressed, for
$k=1,2$, in terms of
\[
L_{k}w=-\sum_{i,j=1}^{N}\frac{\partial }{\partial x_{j}}
(a_{ij}^{k}(x)\frac{\partial w}{\partial x_{i}})+\sum_{i=1}^{N}\frac{
\partial }{\partial x_{j}}(a_{i}^{k}(x)w)
\]
with smooth coefficients $a_{ij}^{k},a_{i}^{k}\in C^1(\overline{\Omega })$.
We also may assume that $b_{k}(x)$ are smooth coefficients on each part of
$\partial \Omega $ and
\begin{equation}
f(u,v)=\mu (u)v, \label{nonlinearity}
\end{equation}
with $\mu (\cdot )$ satisfying the assumptions in \cite{CIR}:
i.e. $\mu \in C[0,+\infty )$, $\mu (0)=0$,
\begin{equation}
0<\mu (z)\leq \overline{\mu }z+\overline{\mu }\quad \text{for any }z>0,
\label{boundedness}
\end{equation}
and such that one of the following properties hold:
\begin{equation}
\mu \text{ is increasing and concave}, \label{A1}
\end{equation}
or
\begin{equation}
\text{there exists $s>0$ such that $\mu$ is increasing on $(0,s)$
and decreasing on }(s,+\infty ). \label{Assum2}
\end{equation}
The $L^1$-framework associated to this system allows to get an improvement
alternative to the main existence and uniqueness result of \cite{crespo2015}
(we do not try to get the more general assumptions on the data but only the
ones which are relevant to our purposes).
\begin{theorem} \label{thm1}
Assume nonnegative the data $u_{0},v_{0}\in L^1(\Omega ),g\in L^{\infty
}(\partial \Omega )$. Then system \eqref{system} admits a weak solution
$u,v\in C([0,T]:L^1(\Omega ))^2$ with $u,v$ being nonnegative functions.
Moreover:
(i) $u_{t}(\cdot ,t),v_{t}(\cdot ,t)\in
L^{\infty }(\Omega )$ for $t\in (0,T]$ and $u,v\in L^2(\delta
,T:H^2(\Omega ))$ for any $\delta >0$,
(ii) there exists a constant $C(T)>0$ such that for $t\in (0,T]$ we have
\begin{gather*}
\| u(\cdot ,t)\| _{L^{\infty }(\Omega )}
\leq \max (t^{-3/2}\| u_{0}\| _{L^1(\Omega
)},\| g\| _{L^{\infty }(\partial \Omega )}) \\
\| v(\cdot ,t)\| _{L^{\infty }(\Omega
)}\leq C(T)t^{-3/2}\| v_{0}\| _{L^1(\Omega )}e^{
\overline{\mu }t},
\end{gather*}
(iii) if, in addition, $\mu (\cdot )$ is Lipschitz continuous
then the weak solution is unique.
\end{theorem}
As a consequence of the above result, in order to define the notion of
asymptotically stable solution (see \cite[Definition 3.1]{CIR}) the
boundedness requirement on the perturbation of the stationary solution is
not any important restriction since for any $t>0$ the solution becomes
bounded (even if the initial data are unbounded).
As a second remark, concerning the paper \cite{CIR}, we point out that in
the special case of the constant (washout) stationary solution
$(S_{1}^{\ast},B_{1}^{\ast })=(1,0)$ the convergence, as
$t\to+\infty $, proved
in \cite[Theorem 3.6]{CIR}, holds in sharper functional spaces (besides to
hold in $L^2(\Omega )$).
\begin{theorem} \label{thm2}
Assume (as in \cite[Theorem 3.6]{CIR}) that
\begin{equation}
\mu (1)<\frac{{\rm Th}_B}{(2{\rm Da} )^2}
+\frac{(\beta _{1}({\rm Da} ,{\rm Th}_B))^2}{{\rm Th}_B}, \label{eq:3_4}
\end{equation}
with $\beta _{1}({\rm Da} ,{\rm Th}_B)$ given in
\cite[Theorem 3.6]{CIR}. Assume $\| u_{0}-1\| _{L^1(\Omega )}$ and $
\| v_{0}\| _{L^1(\Omega )}$ small enough. Then $(u(\cdot,t),v(\cdot ,t))$
converges to $(1,0)$,
strongly in $H^1(\Omega )\cap L^{\infty }(\Omega )$ and weakly in $
H^2(\Omega )$, as $t\to+\infty $.
\end{theorem}
\begin{proof}[Outline of the proof of Theorem \ref{thm1}]
As in \cite{crespo2015}
the results hold by application of a fixed point argument. So, the
qualitative properties mentioned in the statement follow from the
correspondent properties established for the uncoupled systems
\begin{equation}
\begin{gathered}
u_{t}-L_{1}u=F(x,t) \quad \text{in }\Omega \times (0,T), \\
\frac{\partial u}{\partial n}+b_{1}(x)u=g(x) \quad \text{on }\partial \Omega
\times (0,T), \\
u(x,0)=u_{0}(x) \quad \text{on }\Omega ,
\end{gathered}
\end{equation}
and
\begin{equation}
\begin{gathered}
v_{t}-L_{2}v+a(x,t)v=0 \quad \text{in }\Omega \times (0,T), \\
\frac{\partial v}{\partial n}+b_{2}(x)v=0 \quad \text{on }\partial \Omega \times
(0,T), \\
v(x,0)=v_{0}(x) \quad \text{on }\Omega .
\end{gathered}
\end{equation}
The existence, uniqueness and the regularity mentioned in (ii) for solutions
of the uncoupled problem is a consequence of the Semigroup Theory in Banach
Spaces applied to the space $L^1(\Omega )$. The smoothing effect mentioned
in ii) was proved in \cite{Massey} for the case of Dirichlet boundary
conditions by using some properties of the associated Green function
obtained in \cite{Tanabe}. The adaptation to the case of Robin boundary
conditions is a routine matter after the pioneering work by Brezis-Strauss
\cite{Brezis-Strauss} in which more general boundary conditions were
considered (see also the treatment made in \cite{Alt-Luckhaus},
\cite{D-Stakgold}, \cite{Benilan-Petra}).
Notice that the convexity assumption on
$\Omega $ allows the application of the $H^2(\Omega )$-regularity
techniques (see, e.g. \cite{Grisvard}). This assumption could be relaxed but
we do not enter into details here.
The application of the Schauder Fixed Point Theorem is a mimetic
application of the proof given in \cite{crespo2015} for $L^2(\Omega )$
initial data (a $L^1(\Omega )$-compactness argument can be also applied as
in \cite{D-Vrabie}). The regularity $F\in W^{1,1}(0,T:L^{\infty }(\Omega ))$
assumed in \cite{Massey} can be obtained, for the regularity proof of the
fixed point, by means of a previous Steklov average regularization process
following passing to the limit as in \cite{D-Veron}.
The uniqueness of weak solution under the Lipschitz continuity
condition on $\mu (\cdot )$ is an easy task which can be
obtained, for instance, by a simple modification of the proof given in \cite
{crespo2015}. As a matter of fact the uniqueness of solution can be obtained
(as in \cite{D-Stakgold}) when $\mu (\cdot )$ is merely
assumed to be H\"{o}lder continuous and increasing.
The non-negativeness (and even the existence) of solutions for the
coupled system can be also obtained as in \cite{crespo2015} (see also \cite
{d-Hernandez}, \cite{D-Stakgold}, \cite{Padial}, \cite{DGJ} and \cite{Chipot}
for some related works).
\end{proof}
\begin{proof}[Outline of the proof of Theorem \ref{thm2}]
Once we know the $L^2(\Omega )$ convergence (\cite[Theorem 3.6]{CIR}),
we get that, for any $\epsilon >0$,
\[
\max \Big(\int_{\epsilon }^{t}\frac{d}{d\tau }\| u(\tau )\|
_{L^2(\Omega )}^2d\tau ,\int_{\epsilon }^{t}\| \nabla u(\tau
)\| _{L^2(\Omega )}^2d\tau \Big)\leq C,
\]
for some $C>0$ independent of $t$ (see \cite[formula (26)]{CIR}).
Then $u\in W^{1,1}(\epsilon ,+\infty :L^2(\Omega ))\cap L^2(\epsilon ,+\infty
:H^1(\Omega ))$. Moreover, by multiplying by $\frac{\partial }{\partial
\tau }u(\tau )$, it is possible to show that $u\in W^{1,\infty }(\epsilon
,+\infty :L^2(\Omega ))\cap L^{\infty }(\epsilon ,+\infty :H^1(\Omega ))$,
as in \cite[Theorem 6]{D-thellin}. The coercivity of the linear operator
$L_{1}u$ allows to show that in fact $u(t)\to1$ in $H^1(\Omega )$,
as $t\to+\infty $ (see \cite[Theorem 2]{D-thellin}). Notice that
since the $\omega $-limit set for the system is formed by a discrete set of
stationary solutions then we have the convergence
when $t\to +\infty $ and not only for a subsequence
(see \cite[Remark 2]{D-thellin}).
Finally, since the regularity obtained in Theorem 1 and the $L^2(\Omega )$
convergence lead the universal estimate
$\| u\|_{L^2(\delta ,+\infty :H^2(\Omega ))}\leq C$, for some $C>0$
(here $ \delta >0$ is fixed), we get that $u(t)\rightharpoonup 1$ in
$H^2(\Omega ), $ as $t\to+\infty $, and since the inclusion
$H^2(\Omega )\hookrightarrow L^{\infty }(\Omega )$ is compact (remember the
three-dimensional formulation of the problem) we get that $u(t)\to1$
in $L^{\infty }(\Omega )$ as $t\to+\infty $. The proof of the
convergence $v(\cdot ,t)\to0$, strongly in $
H^1(\Omega )\cap L^{\infty }(\Omega )$ and weakly in $H^2(\Omega )$, as $
t\to+\infty $, is entirely similar.
\end{proof}
\subsection*{Acknowledgments}
This research was partially supported by the project
MTM2014-57113-P of the DGISPI (Spain) and the Research Group MOMAT (Ref.
910480) of the UCM.
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\end{document}