J.Bricmont, A.Kupiainen, J. Xin
Global Large Time Self-similarity of a Thermal-Diffusive
Combustion System with Critical Nonlinearity
(53K, LateX)
ABSTRACT. We study the initial value problem of the thermal-diffusive combustion
system: $u_{1,t} = u_{1,x,x} - u_1 u^2_2, u_{2,t} = d u_{2,xx} + u_1 u^2_2,
x \in R^1$, for non-negative spatially decaying initial data of arbitrary
size and for any positive constant $d$. We show that if the initial data
decays to zero sufficiently fast at infinity, then the solution $(u_1,u_2)$
converges to a self-similar solution of the reduced system: $u_{1,t} =
u_{1,xx} - u_1 u^2_2, u_{2,t} = d u_{2,xx}$, in the large time limit. In
particular, $u_1$ decays to zero like ${\cal O}(t^{-\frac{1}{2}-\delta})$,
where $\delta > 0$ is an anomalous exponent depending on the initial data,
and $u_2$ decays to zero with normal rate ${\cal O}(t^{-\frac{1}{2}})$. The
idea of the proof is to combine the a priori estimates for the decay of
global solutions with the renormalization group (RG) method for
establishing
the self-similarity of the solutions in the large time limit.