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: $u_{1,t}=Delt{a}_x{u}_1-u_1{u}_2^m$, $u_{2,t}=dDelt{a}_x{u}_2+u_1{u}_2^m$, $xin{R}^n$, $ngeq1$, $mgeq1$, $d>1$, 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 $dleq1$ 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 $O(loglogt)$ 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.

arXiv comment15 pages, dvi uuencoded