Different asymptotic behavior of global solutions for a parabolic system with nonlinear gradient terms

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:387

Article

Al Elaiw, Abeer²  Tayachi, Slim¹ 

[1] Univ Tunis El Manar, Dept Math, Fac Sci Tunis, Tunis 2092, Tunisia

[2] King Faisal Univ, Dept Math, Fac Sci, Al Hufuf 31982, Saudi Arabia

关键词: Nonlinear parabolic systems;    Global existence;    Nonlinear gradient terms;    Self-similar solutions;    Large time behavior;   

DOI  :  10.1016/j.jmaa.2011.10.014

来源: Elsevier

【 摘 要 】

In this paper we study the global existence and the different asymptotic behavior of mild solutions for the nonlinear parabolic system: partial derivative(t)u = Delta u + a vertical bar del v vertical bar(p), partial derivative(t)v = A v + b vertical bar del u vertical bar(q), t > 0, x is an element of R(N), where a, b is an element of R, N >= 1, 1 < p <= q < 2 and pq > q/N+1+ N+2/N+1. We prove, in particular, that if the initial values behave as u(0, x) similar to omega(1)(x/vertical bar x vertical bar)vertical bar x vertical bar(-alpha). v(0, x) similar to omega(2)(x/vertical bar x vertical bar)vertical bar x vertical bar(-beta) as vertical bar x vertical bar -> infinity, 0 < alpha, beta < N, beta+2-q/q < alpha, alpha+2-P/p < beta and under suitable conditions on omega(1), omega(2), then the resulting solutions are global. Furthermore, although the scaling invariance properties of these initial values and the system are different, we prove that some of the solutions are asymptotic to self-similar solutions of appropriate asymptotic systems which depend on the values of alpha and beta. The asymptotic behavior estimates are given in the W(1.infinity)(R(N)) x W(1.infinity)(R(N))-norm and are stable under some small perturbations. The results of this paper complete those of Al-Elaiw and Tayachi (2010) [1] known only for beta+2-q/q = alpha =a and alpha+2-p/-p = beta. (C) 2011 Elsevier Inc. All rights reserved.

【 授权许可】

Free