# American Institute of Mathematical Sciences

2009, 2009(Special): 11-23. doi: 10.3934/proc.2009.2009.11

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

 1 Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton AB, Canada T6G 2G1 2 Department of Mathematics, University of Alberta, Edmonton, Alberta T6G 2G1 3 Department of Mathematics and Statistics, University of Alberta, University of Alberta, Edmonton, Alberta T6G 2G1

Received  July 2008 Revised  February 2009 Published  September 2009

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].

Citation: Walter Allegretto, Yanping Lin, Zhiyong Zhang. Convergence to convection-diffusion waves for solutions to dissipative nonlinear evolution equations. Conference Publications, 2009, 2009 (Special) : 11-23. doi: 10.3934/proc.2009.2009.11
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