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

2009 ◽  
Vol 22 (3) ◽  
pp. 282-298 ◽  
Author(s):  
Sun Fuqin
Keyword(s):  
Parabolic System ◽  
Global Solutions ◽  
Higher Order ◽  
Semilinear Parabolic System ◽  
Self Similar ◽  
Semilinear Parabolic

ASYMPTOTIC SELF-SIMILAR BEHAVIOR OF SOLUTIONS FOR A SEMILINEAR PARABOLIC SYSTEM

2001 ◽  
Vol 03 (03) ◽  
pp. 363-392 ◽  
Author(s):  
SEIFEDDINE SNOUSSI ◽  
SLIM TAYACHI
Keyword(s):  
Initial Data ◽  
Parabolic System ◽  
Large Time ◽  
Space Dimension ◽  
Mild Solutions ◽  
Small Initial Data ◽  
Semilinear Parabolic System ◽  
Self Similar ◽  
Similar Solutions ◽  
Semilinear Parabolic

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.

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