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

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  • 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].

    Mathematics Subject Classification: Primary: 35H05 ; Secondary: 35K10.


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