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\title{On a nonlinear system arising in a theory of thermal explosion} %
\author{S.H. Rasouli\\
 Department of Mathematics, Faculty of Basic Sciences,
\\Babol Noshirvani University of Technology, Babol, Iran\\e-mail:
s.h.rasouli@nit.ac.ir\\} \setlength{\oddsidemargin}{-0.2in}
\date{}

\begin{document}
\maketitle
\begin{center}
{\bf\large Abstract}\\
\end{center}

The purpose of this paper is to study the existence and multiplicity
of positive solutions for a mathematical model of thermal explosion
which is described by the system
$$
\left\{\begin{array}{ll}
-\Delta u  =  \lambda f(v), & x\in \Omega,\\
-\Delta v  =  \lambda g(u), & x\in \Omega,\\
\mathbf{n}.\nabla u+ a(u) u=0 ,  & x\in\partial \Omega,\\
\mathbf{n}.\nabla v+ b(v) v=0 ,  & x\in\partial \Omega,\\
\end{array}\right.
$$
where $\Omega$ is a bounded smooth domain of $\mathbb{R}^{N},$
$\Delta$ is the Laplacian operator, $\lambda>0$ is a parameter,
$f,g$ belong to a class of non-negative functions that have a
combined sublinear effect at $\infty,$ and $a,b: [0,\infty)
\rightarrow (0,\infty)$ are nondecreasing $C^{1}$ functions. We
establish our existence and multiplicity results by the method of
sub-- and supersolutions.\\\\
\hspace{-0.6 cm}Keywords: Nonlinear system; Thermal explosion; Sub-supersolutions.\\
 %\renewcommand{baselinestretch}{2}
AMS Subject Classification: 35J55, 35J65.
\section{Introduction}
\hspace{0.6 cm}A classical problem in combustion theory is a model
of thermal explosion which occurs due to a spontaneous ignition in a
rapid combustion process. In this paper, we consider a model
involving a nonlinear boundary heat loss which is not a very typical
one in classical combustion theory, but is relevant to some more
applications (see \cite {dafk,nns,ybz,pvg} for details). The model
reads as:
\begin{equation}
\left\{\begin{array}{ll}
\theta(t)-\Delta \theta  =  \lambda f(\eta), & (t,x)\in (0,\infty)\times \Omega,\\
\eta(t)-\Delta \eta  =  \lambda g(\theta), & (t,x)\in (0,\infty)\times \Omega,\\
\mathbf{n}.\nabla \theta+ a(\theta) \theta=0 ,  & (t,x)\in (0,\infty)\times \partial \Omega,\\
\mathbf{n}.\nabla \eta+ b(\eta) \eta=0 ,  & (t,x)\in (0,\infty)\times \partial \Omega,\\
\theta(0,x)= 0 = \eta(0,x).
\end{array}\right.
\end{equation}
Here $\theta,\eta$ are the appropriately scaled temperature in a
bounded smooth domain $\Omega\subset \mathbb{R}^{N,}$ $N\geq 1,$ and
$f,g$ are the normalized reaction rate. We assume that $f,g$
satisfying the following assumptions:
\begin {description}
\item{\bf (H1)} $f,g\in C([0,\infty))$ are nondecreasing functions,
\end {description}
and
\begin {description}
\item{\bf (H2)} $\lim_{s\to\infty}\frac{f(Ag(s))}{s}=0,$ for
all $ A>0.$
\end {description}

On the $C^{2}$ boundary $\partial \Omega,$ with the outward unit
normal denoted by $\mathbf{n}$, the heat-loss parameters
$a(\theta),b(\eta)$ are assumed to satisfy the following hypothesis:
\begin {description}
\item{\bf (H3)}  $a,b : [0,\infty)\rightarrow(0,\infty)$ are nondecreasing bounded $C^{1}$ functions.
\end {description}

Physically this assumption means that a heat loss through the
boundary always exists and increases linearly with the temperature
even in the small temperature regime.\\

A bifurcation (or scaling) parameter $\lambda > 0$ can be associated
with the size of domain $\Omega$ in $(1)$ which grows linearly as
the measure of $\Omega$ increases. It is well known that, after
normalizing for the size of $\Omega$, the long term behavior of
solution of $(1)$ is close to the solution of the time-independent
system:
\begin{equation}
\left\{\begin{array}{ll}
-\Delta u  =  \lambda f(v), & x\in \Omega,\\
-\Delta v  =  \lambda g(u), & x\in \Omega,\\
\mathbf{n}.\nabla u+ a(u) u=0 ,  & x\in\partial \Omega,\\
\mathbf{n}.\nabla v+ b(v) v=0 ,  & x\in\partial \Omega.\\
\end{array}\right.
\end{equation}

The motivation for this study cames from the work in \cite{ek} where
the authors established the existence, uniqueness and multiplicity
of positive solutions for certain range of $\lambda$ for the single
equation of the form
$$
\left\{\begin{array}{ll}
-\Delta u  =  \lambda f(u), & x\in \Omega,\\
\mathbf{n}.\nabla u+ a(u) u=0 ,  & x\in\partial \Omega.\\
\end{array}\right.
$$

Here we extend this study to Laplacian system of the form $(1).$ In
\cite{ja}, Ali-Shivaji-Ramaswamy discussed the existence of multiple
positive solutions to such systems with Dirichlet boundary
conditions. One can refer to \cite{db,ek1} for some recent existence
and uniqueness results of elliptic problems with nonlinear boundary
conditions.
\section{Existence results}
\hspace{0.6 cm}In this section, we shall establish our existence
results via the method of sub - supersolution. A pair of nonnegative
functions $(\psi_{1},\psi_{2})\in W^{1,2}\cap
C(\overline{\Omega})\times  W^{1,2}\cap C(\overline{\Omega})$ and
$(z_{1},z_{2})\in W^{1,2}\cap C(\overline{\Omega})\times W^{1,2}\cap
C(\overline{\Omega})$ are called a subsolution and supersolution of
(1) if they satisfy
\begin{equation}
\left\{\begin{array}{ll}
-\Delta \psi_{1} \leq \lambda f(\psi_{2}), & x\in \Omega,\\
-\Delta \psi_{2}  \leq  \lambda g(\psi_{1}), & x\in \Omega,\\
\mathbf{n}.\nabla \psi_{1}+ a(\psi_{1}) \psi_{1}\leq0 ,  & x\in\partial \Omega,\\
\mathbf{n}.\nabla \psi_{2}+ b(\psi_{2}) \psi_{2}\leq0 ,  & x\in\partial \Omega,\\
\end{array}\right.
\end{equation}
and
\begin{equation}
\left\{\begin{array}{ll}
-\Delta z_{1} \geq \lambda f(z_{2}), & x\in \Omega,\\
-\Delta z_{2}  \geq  \lambda g(z_{1}), & x\in \Omega,\\
\mathbf{n}.\nabla z_{1}+ a(z_{1}) z_{1}\geq 0 ,  & x\in\partial \Omega,\\
\mathbf{n}.\nabla z_{2}+ b(z_{2}) z_{2}\geq0 ,  & x\in\partial \Omega,\\
\end{array}\right.
\end{equation}
respectively. It is well known that if there exist sub and
supersolutions $(\psi_{1},\psi_{2})$ and $(z_{1},z_{2})$
respectively of $(1)$ such that $(\psi_{1},\psi_{2})\leq
(z_{1},z_{2}) . $ Then $(1)$ has a solution $ (u,v)$ such that
$(u,v)$
$\in[(\psi_{1},\psi_{2}),(z_{1},z_{2})]$  ( see \cite {ha,fi} ).\\

By strict sub and super-solutions we understand functions
$(\psi_{1},\psi_{2})$ and $(z_{1},z_{2})$ for which strict
inequalities $(3)$ and $(4)$ hold.\\

Our multiplicity results are obtained by constructing sub and
super-solution pairs that satisfy the following Lemma:\\\\
{\bf Lemma 2.1.} ( See \cite{fi,cm,rs} ). Suppose the system $(1)$
has a sub-solution $( \psi_{1},\psi_{2}),$ a strict super-solution
$(\zeta_{1},\zeta_{2}),$ a strict sub-solution $(w_{1},w_{2}),$ and
a super-solution $(z_{1},z_{2})$ for $(1)$ such that
$$
( \psi_{1},\psi_{2})\leq (\zeta_{1},\zeta_{2})\leq (z_{1},z_{2}),
$$
$$
( \psi_{1},\psi_{2})\leq (w_{1},w_{2})\leq (z_{1},z_{2}),
$$
and $(w_{1},w_{2})\nleq (\zeta_{1},\zeta_{2}).$ Then $(1)$ has at
least three distinct solutions $(u_{i},v_{i}),$ $i = 1, 2, 3$ such
that
$$
 (u_{1}, v_{1})\in [( \psi_{1},\psi_{2}), (\zeta_{1},\zeta_{2})],\,\,\, (u_{2}, v_{2})\in [( w_{1},w_{2}), (z_{1},z_{2})]
$$
and
$$
(u_{3}, v_{3}) \in \Big[( \psi_{1},\psi_{2}),
(z_{1},z_{2})\Big]\backslash \Big(\Big[( \psi_{1},\psi_{2}),
(\zeta_{1},\zeta_{2})\Big]\cup \Big[ ( w_{1},w_{2}),
(z_{1},z_{2})\Big] \Big).\\
$$

To precisely state our existence result we consider the unique
classical solution $e_{r}$ of the following linear elliptic problem
\begin{equation}
\left\{\begin{array}{ll}
-\Delta e_{r}  = 1, & x\in \Omega,\\
\mathbf{n}.\nabla e_{r}+ r_{0} e_{r}= 0 ,  & x\in\partial \Omega,\\
\end{array}\right.
\end{equation}
for $r=a,b,$ where $r_{0}=r(0).$ Then we establish:\\\\
{\bf Theorem 2.2.} Let $(H1)-(H3)$ hold and $f(0)$ or $g(0)$ be
strictly positive. Then $(1)$ has a
positive solution $(u,v)$ for all $\lambda>0.$\\\\
{\bf Proof.} It is easy to see that $( \psi_{1},\psi_{2})= (0,0)$ is
a subsolution of $(1).$ We now construct the supersolution
$(z_{1},z_{2}).$ Let $(z_{1},z_{2})= \Big(C_{\lambda}e_{a}, \lambda
g(C_{\lambda}\|e_{b}\|_{\infty})e_{b}\Big),$ where $C_{\lambda}$ is
a large number to be chosen later. We shall verify that
$(z_{1},z_{2})$ is a supersolution of $(1)$ for all $\lambda>0.$ By
$(H2)$ we can choose $C_{\lambda}$ large enough so that
$$
C_{\lambda}\geq \lambda f\Big( \lambda
g(C_{\lambda}\|e_{b}\|_{\infty})\|e_{b}\|_{\infty}\Big),
$$
and therefore
\begin{eqnarray*}
\begin{split}
-\Delta z_{1}=C_{\lambda}&\geq \lambda f\Big( \lambda
g(C_{\lambda}\|e_{b}\|_{\infty})\|e_{b}\|_{\infty}\Big)\\
&\geq \lambda f\Big( \lambda g(C_{\lambda}\|e_{b}\|_{\infty})e_{b}\Big)\\
&= \lambda f(z_{2})\,\,\,\text{in}\,\Omega,
\end{split}
\end{eqnarray*}
and
\begin{eqnarray*}
\begin{split}
\mathbf{n}.\nabla z_{1}+ a(z_{1}) z_{1}&\geq
C_{\lambda}\mathbf{n}.\nabla
e_{a}+C_{\lambda}e_{a}a_{0}\\
&= C_{\lambda}(\mathbf{n}.\nabla
e_{a}+e_{a}a_{0})\\
&=0\,\,\,\text{on}\,\partial\Omega.\\
\end{split}
\end{eqnarray*}

Next,
\begin{eqnarray*}
\begin{split}
-\Delta z_{2}&=\lambda g\Big(C_{\lambda}\|e_{b}\|_{\infty}\Big)\\
&\geq \lambda g\Big(C_{\lambda} e_{b}\Big)\\
&= \lambda g(z_{1})\,\,\,\text{in}\,\Omega,
\end{split}
\end{eqnarray*}
and
\begin{eqnarray*}
\begin{split}
\mathbf{n}.\nabla z_{2}+ b(z_{2}) z_{2}&\geq\lambda
g\Big(C_{\lambda}\|e_{b}\|_{\infty}\Big)\mathbf{n}.\nabla
e_{b}+\lambda
g\Big(C_{\lambda}\|e_{b}\|_{\infty}\Big)e_{b}b_{0}\\
&=\lambda g\Big(C_{\lambda}\|e_{b}\|_{\infty}\Big)(\mathbf{n}.\nabla
e_{b}+ b_{0} e_{b})\\
&=0\,\,\,\text{on}\,\partial\Omega,
\end{split}
\end{eqnarray*}

 which implies that $(z_{1},z_{2})$ is indeed a positive supersolution of $(1)$. Therefore
 $(1)$ has a positive solution for all $\lambda>0.$\hspace{1
 cm}$\Box$\\

 Our second result concerns with multiplicity of solution for the
 system $(1)$ and gives an estimate on the parameter $\lambda$ when
 such a situation occurs. For positive constants $a_{i},b_{i};$
 $i=1,2,$ define
 $$
Q_{1}(a_{1},b_{1})= \min
\{\frac{a_{1}}{f(b_{1})},\frac{b_{1}}{g(a_{1})}\}
 $$
 and
  $$
Q_{2}(a_{2},b_{2})= \max
\{\frac{a_{2}}{f(b_{2})},\frac{b_{2}}{g(a_{2})}\}.\\
 $$

 Then we establish:\\\\
{\bf Theorem 2.3.} Assume $f(0)$ or $g(0)$ be strictly positive. Let
$B_{R}$ be the largest ball of radius $R$ inscribed in $\Omega,$ for
$0<\epsilon<R,$ we define
$$
C_{1}(\Omega)= \inf_{\epsilon}
\frac{N}{\epsilon^{N}}\frac{R^{N-1}}{R-\epsilon},
$$
and $C(\Omega)= C_{1}(\Omega) \|e_{r}\|_{\infty},$ for $r=a,b.$ Let
$(H1)-(H3)$ hold and $\frac{Q_{1}}{Q_{2}}>C(\Omega)$ for some
$a_{i},b_{i},$ $i=1,2.$ Then $(1)$ has at least three positive
solutions for $\lambda \in (\lambda_{*},\lambda^{*}),$ where
$\lambda_{*}=CQ_{2}$ and $\lambda^{*}=\frac{Q_{1}}{\|e_{r}\|_{\infty}},$ for $r=a,b.$\\\\
{\bf Proof.} We will establish a pair of subsolutions $(
\psi_{1},\psi_{2}),$ $(w_{1}, w_{2})$ and a pair of supersolutions
$( \zeta_{1},\zeta_{2}),$ $( z_{1},z_{2}),$ satisfying Lemma $2.1.$
Clearly $( \psi_{1},\psi_{2}) = (0, 0)$ is a subsolution of $(1).$\\

We next construct a positive supersolution $( \zeta_{1},\zeta_{2}),$
of $(1)$ when $\lambda<\frac{Q_{1}}{\|e_{r}\|_{\infty}},$ for
$r=a,b.$ Since $\lambda<\frac{a_{1}}{f(b_{1})\|e_{a}\|_{\infty}},$
we can choose $\epsilon>0$ so small that $\lambda f(b_{1})<
\frac{a_{1}}{\epsilon+\|e_{a}\|_{\infty}}.$ Let $(
\zeta_{1},\zeta_{2})=
(a_{1}\frac{e_{a}+\epsilon}{\|e_{a}\|_{\infty}+\epsilon},b_{1}\frac{e_{b}+\epsilon}{\|e_{b}\|_{\infty}+\epsilon}).$
Then, we have
\begin{eqnarray*}
\begin{split}
-\Delta \zeta_{1}=\frac{a_{1}}{\epsilon+\|e_{a}\|_{\infty}}&>
\lambda f(b_{1})\\
&\geq \lambda f\Big(b_{1}\frac{e_{b}+\epsilon}{\|e_{b}\|_{\infty}+\epsilon}\Big)\\
&= \lambda f(\zeta_{2})\,\,\,\text{in}\,\Omega,
\end{split}
\end{eqnarray*}
and
\begin{eqnarray*}
\begin{split}
\mathbf{n}.\nabla \zeta_{1}+ a(\zeta_{1}) \zeta_{1}&\geq
\frac{a_{1}}{\epsilon+\|e_{a}\|_{\infty}}\Big( \mathbf{n}.\nabla
e_{a}+(e_{a}+\epsilon)a_{0}\Big)\\
&=\frac{a_{1}}{\epsilon+\|e_{a}\|_{\infty}}( \mathbf{n}.\nabla
e_{a}+a_{0} e_{a}+a_{0}\epsilon)\\
&=\frac{a_{1}a_{0}\epsilon}{\epsilon+\|e_{a}\|_{\infty}}\\
&>0\,\,\,\text{on}\,\partial\Omega.\\
\end{split}
\end{eqnarray*}
Similar argument shows that $\zeta_{2}$ satisfies $-\Delta
\zeta_{2}> \lambda g(\zeta_{1})$\,in\,$\Omega,$ and
$\mathbf{n}.\nabla \zeta_{2}+ b(\zeta_{2}) \zeta_{2}> 0.$\\

Next let us construct a strict sub-solution $(w_{1},w_{2})$ of
$(1).$ First note that a system
$$
\left\{\begin{array}{ll}
-\Delta u_{D}  =  \lambda f(v_{D}), & x\in \Omega,\\
-\Delta v_{D}  =  \lambda g(u_{D}), & x\in \Omega,\\
u_{D}=0=v_{D} ,  & x\in\partial \Omega,\\
\end{array}\right.
$$
admits a strict sub-solution $(w_{1D},w_{2D})$ with
$\|w_{1D}\|_{\infty}\geq a_{2}$ and $\|w_{2D}\|_{\infty}\geq b_{2}$
provided $\lambda<\lambda^{*}$ (see \cite{ja}). Then we have
$(w_{1},w_{2})\nleq (\zeta_{1},\zeta_{2}).$ By the Hopf's lemma we
have that $\mathbf{n}.\nabla w_{iD} < 0$ for $i=1,2.$ Therefore,
setting $w_{1}=w_{1D}$ and $w_{2}=w_{2D}$ we obtain a strict
sub-solution for $(1)$ for $\lambda>\lambda_{*}.$\\

Let $(z_{1},z_{2})$ be the super solution as in the proof of Theorem
$2.2$ Further $ w_{i},\zeta_{i}\leq z_{i},$ $i=1,2$ for
$C_{\lambda}$ large. Hence there exist positive solutions
$(u_{i},v_{i}),$ $i = 1, 2, 3$ such that
$$
 (u_{1}, v_{1})\in [( \psi_{1},\psi_{2}), (\zeta_{1},\zeta_{2})],\,\,\, (u_{2}, v_{2})\in [( w_{1},w_{2}), (z_{1},z_{2})]
$$
and
$$
(u_{3}, v_{3}) \in \Big[( \psi_{1},\psi_{2}),
(z_{1},z_{2})\Big]\backslash \Big(\Big[( \psi_{1},\psi_{2}),
(\zeta_{1},\zeta_{2})\Big]\cup \Big[ ( w_{1},w_{2}),
(z_{1},z_{2})\Big] \Big).\\\hspace{1
 cm}\Box\\
$$
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\end{thebibliography}
\end{document}\\