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

We investigate the asymptotic profile to the Cauchy problem for a non-linear dissipative evolution system with conservational form {ψt = - (1 - α)ψ - θx + αψxx θt = - (1 - α)θ + v 2ψx + (ψθ)x + αθxx provided that the initial data are small, where constants α, v are positive satisfying v2<4α(1 - α), α<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 → ∞, the solution ψ → De-(1-α-v2/4α)t G(t, x)cos((v/2α)x + π/4 + β) and solution θ → -Dve-(1-α-v2/4α)t G(t, x) sin((v/2α)x + π/4 + β) in the Lp sense, where G(t, x) denotes the heat kernel and D = 2√2(+2 + -2) is determined by the initial data and the solution to a reformulated problem obtained in Section 3, β is related to + and - 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: 604-614).