Asymptotic behavior of solution to nonlinear evolution equations with damping

In this paper, we consider the global existence and the asymptotic behavior of solutions to the Cauchy problem for the following nonlinear evolution equations with ellipticity and dissipative effects ψt =−(1−α)ψ − θx +αψxx, (t, x) ∈ (0,∞)× R, θt =−(1 −α)θ +νψx +2ψθx +αθxx, (E) with initial data (ψ, θ)(x, 0) = ψ0(x), θ0(x) →(ψ±, θ±) as x→±∞, (I) where α and ν are positive constants such that α < 1, ν < 4α(1 − α). Under the assumption that |ψ+ − ψ−| + |θ+ − θ−| is sufficiently small, we show that if the initial data is a small perturbation of the parabolic system defined by (2.4) which are obtained by the convection–diffusion equations (2.1), and solutions to Cauchy problem (E) and (I) tend asymptotically to the convection–diffusion system with exponential rates. Precisely speaking, we derive the asymptotic profile of (E) by Gauss kernel G(t, x) as follows: ψ θ − ν(ψ+ − ψ−)2 + (θ+ − θ−)2e−(1−α− ν 4α )t × x −∞ G(y, t +1) 1 √ν sin( √ν 2α y + β¯0) cos( √ν 2α y + β¯0) dy −e−(1−α)t φ+ θ+ −2e−(1−α− ν 4α )t × R G(t, y) · cos( √ν 2α y), 1 √ν sin( √ν 2α y) −√ν sin( √ν 2α y), cos( √ν 2α y) · u0(x− y) v0(x − y) dy Lp (Rx) = e−(1−α− ν 4α )t O (1 +t)−(1− 1 p ) . The same problem was studied by Tang and Zhao [S.Q. Tang, H.J. Zhao, Nonlinear stability for dissipative nonlinear evolution equations with ellipticity, J. Math. Anal. Appl. 233 (1999) 336–358], Nishihara [K. Nishihara, Asymptotic profile of solutions to nonlinear dissipative evolution system with ellipticity, Z. Angew. Math. Phys. 57 (4) (2006) 604–614] for the case of (ψ±, θ±) = (0, 0).