Convergence to diffusion waves for nonlinear evolution equations with different end states

In this paper, we consider the global existence and the asymptotic decay of solutions to the Cauchy problem for the following nonlinear evolution equations with ellipticity and dissipative effects: ψt =−(1−α)ψ +ψψx + f (θ) x +αψxx, θt =−(1−α)θ +νψx + (ψθ)x +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, sν < 4α(1 − α) (s is defined in (1.14)). Under the assumption that |ψ+ − ψ−| + |θ+ −θ−| is sufficiently small, we show that if the initial data is a small perturbation of the diffusion waves defined by (2.5) which are obtained by the diffusion equations (2.1), solutions to Cauchy problem (E) and (I) tend asymptotically to those diffusion waves with exponential rates. The analysis is based on the energy method. The similar 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] for the case of (ψ±, θ±)