Asymptotic self-similar behavior of solutions for a semilinear parabolic system

TLDR: In this paper, the authors studied the global existence and the asymptotic self-similar behavior of solutions of the semilinear parabolic system ∂tu=Δu+a1|u|p1-1u+b1|v|q 1-1v, ∂tv =Δv+a2|v,p2-1 v+b2|u,u|q 2-1 u|q2-u,q 2 -1u, q 2 1u,

Abstract: This paper studies the global existence and the asymptotic self-similar behavior of solutions of the semilinear parabolic system ∂tu=Δu+a1|u|p1-1u+b1|v|q1-1v, ∂tv=Δv+a2|v|p2-1v+b2|u|q2-1u, on (0,∞)×ℝn, where a1, bi∈ℝ and pi, qi>1. Let p=min{p1,p2, q1(1+q2)/(1+q1), q2(1+q1)/(1+q2)}. Under the condition p>1+2/n we prove the existence of globally decaying mild solutions with small initial data. Some of them are asymptotic, for large time, to self-similar solutions of appropriate asymptotic systems having each one a self-similar structure. All possible asymptotic self-similar behaviors are discussed in terms of exponents pi, qi, the space dimension n and the asymptotic spatial profile of the related initial data.