Existence of global solutions for systems of reaction-diffusion equations on unbounded domains - Salah Badraoui (Electronic Journal of Differential Equations, 2002)
We consider, an initial-value problem for the thermal-diffusive combustion system $$displaylines{ u_t=aDelta u-uh(v) cr v_t=bDelta u+dDelta v+uh(v), }$$ where $a>0$, $d>0$, $beq 0$, $xin mathbbR^n$, $ngeq 1$, 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 $b=0$, $a=1$, $dleq 1$ and $h(v)=v^m$ with $min mathbbN^*$, the solutions are uniformly bounded in time. When $d>a=1$, $b=0$, $h(v)=v^m$ with $min mathbbN^*$, 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 $O(loglog t)$ 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 $v_0geq frac ba-du_0$.