Asymptotic Self-Similar Behavior of Solutions for a Semilinear Parabolic System

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.

Asymptotically self-similar global solutions of a general semilinear heat equation

“…Finally, we mention that results similar to those in the present article have been obtained for nonlinear Schrödinger equations in [5] and [29] and for parabolic systems in [26].…”

“…The only previous result we know concerning solutions of (1.2) with a = 1 which are asymptotic to self-similar solutions whose profiles decay slowly are due to Cazenave and the third author [3]. Since then, the ideas in these papers have been further developed and applied to several different semilinear problems: to the Navier-Stokes system [8,21], to nonlinear heat equations [3,22,27], to the nonlinear Schrödinger equation [3][4][5]7,18,20,23], to the nonlinear wave equation [19,24] and to semilinear parabolic systems [25,26]. See Theorem 6.2 in [3].…”

Life span of solutions of a weakly coupled parabolic system

“…Then there exists ε > 0 such that w is global whenever ( u 0 r1 + v 0 r2 ) ≤ ε, see [9]. A related result, showing that fast decaying (in space) solutions are global, was obtained by Snoussi and Tayachi in [23].…”

Large time behavior of solutions for a complex-valued quadratic heat equation

“…Many results have been established concerning existence of self-similar solutions and asymptotically self-similar solutions for other systems using the methods used in [1,8]. In [6], the semi-linear parabolic system with nonlinear terms of the form a j |u| pj −1 u + b j |v| qj −1 v, p j , q j > 1, a j , b j ∈ R, j = 1, 2, have been studied. The global existence and the existence of asymptotically self-similar solution are shown under some conditions on the parameters.…”