Abstract
We investigate the asymptotic profile to the Cauchy problem for a nonβlinear dissipative evolution system with conservational form
provided that the initial data are small, where constants Ξ±, Ξ½ are positive satisfying Ξ½^2^<4Ξ±(1 β Ξ±), Ξ±<1. In (J. Phys. A 2005; 38:10955β10969), 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 β β the solution $ {\psi \rightarrow De^{-(1-\alpha-\nu^{2}/{4 \alpha})t}G(t, x) \cos ((\nu / 2{\alpha})x + {\Pi / 4} + \beta)}$ and solution ${\theta \rightarrow - D \nu e^{-(1-\alpha-\nu^{2}/{ 4\alpha}) t}G(t, x)\sin((\nu/2{\alpha})x + {\Pi / 4} + \beta)}$ in the L^p^ sense, where G(t, x) denotes the heat kernel and $D=2\sqrt{2(\vartheta_{+}^{2}+\vartheta_{-}^{2})}$ is determined by the initial data and the solution to a reformulated problem obtained in Section 3, Ξ² is related to Ο~+~ and Ο~β~ 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. Z. Angew Math Phys 2006; 57: 604β614). Copyright Β© 2007 John Wiley & Sons, Ltd.