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

We study the initial value problem of the thermal-diffusive combustion systemu1, t=u1, x, x−u1u22,u2, t=du2, xx+u1u22,x R1, for non-negative spatially decaying initial data of arbitrary size and for any positive constantd. We show that if the initial data decay to zero sufficiently fast at infinity, then the solution (u1, u2) converges to a self-similar solution of the reduced systemu1, t=u1, xx−u1u22,u2, t=du2, xx, in the large time limit. In particular,u1decays to zero like (t−1/2−δ), whereδ>0 is an anomalous exponent depending on the initial data, andu2decays to zero with normal rate (t−1/2). The idea of the proof is to combine the a priori estimates for the decay of global solutions with the renormalization group method for establishing the self-similarity of the solutions in the large time limit.