Electronic Journal of Differential Equations,
Vol. 2002(2002), No. 74, pp. 1-10.
Title: Existence of global solutions for systems of reaction-diffusion
equations on unbounded domains
Author: Salah Badraoui (Univ. du 8 Mai 1945-Guelma, Algeria)
Abstract:
We consider, an initial-value problem for the thermal-diffusive
combustion system
$$\displaylines{
u_t=a\Delta u-uh(v) \cr
v_t=b\Delta u+d\Delta v+uh(v),
}$$
where $a>0$, $d>0$, $b\neq 0$, $x\in \mathbb{R}^n$, $n\geq 1$,
with $h(v)=v^m$, $m$ is an even nonnegative integer, and the
initial data $u_0$, $v_0$ are bounded uniformly continuous and
nonnegative. It is known that by a simple comparison if
$b=0$, $a=1$, $d\leq 1$ and $h(v)=v^m$ with $m\in \mathbb{N}^*$,
the solutions are uniformly bounded in time.
When $d>a=1$, $b=0$, $h(v)=v^m$ with $m\in \mathbb{N}^*$,
Collet and Xin [2] proved the existence of global classical
solutions and showed that the $L^\infty $ norm of $v$ can not
grow faster than $O(\log\log t)$ for any space dimension.
In our case, no comparison principle seems to apply.
Nevertheless using techniques form [2], we essentially prove
the existence of global classical solutions if $a