Asymptotic profile of solutions to a non-linear dissipative evolution system with conservation

作者：Wang Zhian^*; Sang Hong

来源：Mathematical Methods in the Applied Sciences, 2007, 30(8): 977-994.

DOI：10.1002/mma.830

• asymptotic profile
• global existence
• a priori estimates
• numerical simulations

We investigate the asymptotic profile to the Cauchy problem for a non-linear dissipative evolution system with conservational form
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provided that the initial data are small, where constants alpha, v are positive satisfying v(2) < 4 alpha(1 - alpha), alpha < 1. In (J. Phys. A 2005; 38:10955-10969), the global existence and optimal decay rates of the solution to this problem have been obtained. The aim of this paper is to apply the heat kernel to examine more precise behaviour of the solution by finding out the asymptotic profile. Precisely speaking, we show that, when time t -> infinity, the solution psi -> De(-(1-alpha-v2/4 alpha)t) G (t, x) cos((v/2 alpha)x + pi/4 + beta) and solution 0 -> -Dve(-(1-alpha-v2/4 alpha)t) G(t, x) sin((v/2 alpha)x + pi/4 + beta) in the L-p sense, where G(t, x) denotes the heat kernel and D = 2 root(2(v(+)(2) + v(-)(2))) is determined by the initial data and the solution to a reformulated problem obtained in Section 3, beta is related to v(+) and v(-) which are determined by (41) in Section 4. The numerical simulation is presented in the end. The motivation of this work thanks to Nishihara (Asymptotic profile of solutions to nonlinear dissipative evolution system with ellipticity.

• 出版日期2007-5-25