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

TL;DR

This paper proves the global existence of solutions and bounds their large-time growth for thermal-diffusive combustion systems on R^n, even in cases lacking comparison principles, using local estimates and decaying test functions.

Contribution

It establishes the existence of global classical solutions and bounds the growth of solutions in systems with thermal-diffusive reactions where traditional comparison methods fail.

Findings

Solutions are globally bounded in time.

The $L^{ abla} $ norm of $u_2$ grows at most logarithmically double-logarithmically.

Results apply to Arrhenius type reactions.

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.