Global Large Time Self-similarity of a Thermal-Diffusive Combustion System with Critical Nonlinearity

Details Global Large Time Self-similarity of a Thermal-Diffusive Combustion System with Critical Nonlinearity Global Large Time Self-similarity of a Thermal-Diffusive Combustion System with Critical Nonlinearity

1995-01-25

arxiv.org/abs/chao-dyn/9501012v1

Nonlinear Sciences

Chaotic Dynamics

22pages, Latex, bricmont@fyma.ucl.ac.be,ajkupiai@cc.helsinki.fi, xin@math.arizona.edu

Scientific paper

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.

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