Electron. J. Diff. Eqns., Vol. 2002(2002), No. 74, pp. 1-10.

Existence of global solutions for systems of reaction-diffusion equations on unbounded domains

Salah Badraoui

We consider, an initial-value problem for the thermal-diffusive combustion system
   u_t=a\Delta u-uh(v)  \cr
   v_t=b\Delta u+d\Delta v+uh(v),
where $a positive, $d positive$, $b\neq 0$, $x\in \mathbb{R}^n$, $n\geq 1$, 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 $d greater than a=1$, $b=0$, $h(v)=v^m$ 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 $a less than d$, $b less than 0$, 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.

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Salah Badraoui
Universite du 8 Mai 1945-Guelma,
Faculte des Sciences et Technologie,
Laboratoire LAIG,
BP.401, Guelma 24000, Algeria
email: s_badraoui@hotmail.com

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