Dynamics And Universality Of An Isothermal Combustion Problem In 2D

In this paper, the Cauchy problem of the system u1,t = Δu1 - u1u2m, u2,t = dΔu2 + u1u2m is studied, where x ∈ R2, m ≥ 1 and d \u3e 0 is the Lewis number. This system models isothermal combustion (see [7]), and auto-catalytic chemical reaction. We show the global existence and regularity of solutions with non-negative initial values having mild decay as |x| → ∞. More importantly, we establish the exact spatio-temporal profiles for such solutions. In particular, we prove that for m = 1, the exact large time behavior of solutions is characterized by a universal, non-Gaussian spatio-temporal profile, with anomalous exponents, due to the fact that quadratic nonlinearity is critical in 2D. Our approach is a combination of iteration using Renormalization Group method, which has been developed into a very powerful tool in the study of nonlinear PDEs largely by the pioneering works of Bricmont, Kupiainen and Lin [6], Bricmont, Kupiainen and Xin, [7], (see also [9]) and key estimates using the PDE method. © World Scientific Publishing Company