Convergence to convection-diffusion waves for solutions to dissipative nonlinear evolution equations

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 damping $$ \left\{\begin{array}{l} \psi_t = -(1-\alpha) \psi - \theta_x + \alpha \psi_{x x} + \psi\psi_x,                                                                                                            (E)\\ \theta_t = -(1-\alpha)\theta + \nu \psi_x + 2\psi\theta_x + \alpha \theta_{x x}, \end{array} \right. $$ with initial data converging to different constant states at infinity $$(\psi,\theta)(x,0)=(\psi_0(x), \theta_0(x)) \rightarrow (\psi_{\pm}, \theta_{\pm}) \ \ {as} \ \ x \rightarrow \pm \infty,                                                             (I) $$ where $\alpha$ and $\nu$ are positive constants such that $\alpha <1$, $\nu <4\alpha(1-\alpha)$. Under the assumption that $|\psi_+ - \psi_- |+| \theta_+ - \theta_-|$ is sufficiently small, we show that if the initial data is a small perturbation of the convection-diffusion waves defined by (11) which are obtained by the parabolic system (9), solutions to Cauchy problem (E) and (I) tend asymptotically to those convection-diffusion waves with exponential rates. We mainly propose a better asymptotic profile than that in the previous work by [13,3], and derive its decay rates by weighted energy method instead of considering the linearized structure as in [3].