Asymptotics of dissipative nonlinear evolution equations with ellipticity: different end states

In this paper, we consider the global existence and asymptotic behaviors of solutions to the Cauchy problem for the following nonlinear evolution equations with ellipticity and dissipative effects: (E){ψt=−(1−α)ψ−θx+αψxx,θt=−(1−α)θ+νψx+2ψθx+αθxx, with initial data (I)(ψ,θ)(x,0)=(ψ0(x),θ0(x))→(ψ±,θ±)as x→±∞, 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 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 same problem was studied by Tang and Zhao [J. Math. Anal. Appl. 233 (1999) 336-358] for the case of (ψ±,θ±)=(0,0).