Global Existence and Large Time Asymptotic Bounds of $L^{infty}$ Solutions of Thermal Diffusive Combustion Systems on $R^{n}$

Abstract

We consider the initial value problem for the thermal-diffusive combustion systems of the form: $u_{1,t}= Delta_{x}u_1 - u_1 u_2^m$, $u_{2,t}= d Delta_{x} u_2 + u_1 u_2^m$, $x in R^{n}$, $n geq 1$, $m geq 1$, $d > 1$, with bounded uniformly continuous nonnegative initial data. For such initial data, solutions can be simple traveling fronts or complicated domain walls. Due to the well-known thermal-diffusive instabilities when $d$, the Lewis number, is sufficiently away from one, front solutions are potentially chaotic. It is known in the literature that solutions are uniformly bounded in time in case $d leq 1$ by a simple comparison argument. In case $d >1$, no comparison principle seems to apply. Nevertheless, we prove the existence of global classical solutions and show that the $L^{infty}$ norm of $u_2$ can not grow faster than $O(log log t)$ for any space dimension. Our main tools are local $L^{p}$ a-priori estimates and time dependent spatially decaying test functions. Our results also hold for the Arrhenius type reactions.