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  $a<d$, $b<0$,
  and  $v_0\geq \frac b{a-d}u_0$.
   
Submitted December 5, 2001. Published August 19, 2002.
Math Subject Classifications: 35B40, 35B45, 35K55, 35K65.
Key Words: Reaction-diffusion systems; positivity; 
          global existence; boundedness.