  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.
