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

Pages: 11 - 23, Issue Special, September 2009

 Abstract        Full Text (197.0K)              

Walter Allegretto - Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton AB, Canada T6G 2G1, Canada (email)
Yanping Lin - Department of Mathematics, University of Alberta, Edmonton, Alberta T6G 2G1, Canada (email)
Zhiyong Zhang - Department of Mathematics and Statistics, University of Alberta, University of Alberta, Edmonton, Alberta T6G 2G1, Canada (email)

Abstract: 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].

Keywords:  Evolution equations, diffusion waves, decay rate, energy method, a priori estimates
Mathematics Subject Classification:  Primary: 35H05 ; Secondary: 35K10

Received: July 2008;      Revised: February 2009;      Published: September 2009.