Asymptotically Self-similar Global Solutions for a Higher-order Semilinear Parabolic System

Abstract

In this paper, we study the higher-order semilinear parabolic system u_t+(-Δ)^mu=a|v|^{p-1}v, t(x)∈R^1_+×R^N, v_t+(-Δ)^mv=b|u|^{q-1}u, t(x)∈R^1_+×R^N, u(0,x)=φ(x), v(0,x)=ψ(x), x∈R^N, where m, p, q > 1, a,b∈R. We prove that the global existence of mild solutions for small initial data with respect to certain norms. Some of these solutions are proved to be asymptotically self-similar.