# American Institute of Mathematical Sciences

December  2012, 32(12): 4069-4110. doi: 10.3934/dcds.2012.32.4069

## Dafermos regularization of a diffusive-dispersive equation with cubic flux

 1 Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205 2 Department of Mathematics, Shepherd University, Shepherdstown, WV 25443-5000, United States

Received  June 2011 Revised  June 2012 Published  August 2012

We study existence and spectral stability of stationary solutions of the Dafermos regularization of a much-studied diffusive-dispersive equation with cubic flux. Our study includes stationary solutions that corresponds to Riemann solutions consisting of an undercompressive shock wave followed by a compressive shock wave. We use geometric singular perturbation theory (1) to construct the solutions, and (2) to show that asmptotically, there are no large eigenvalues, and any order-one eigenvalues must be near $-1$ or a certain number $\lambda^*$. We give numerical evidence that $\lambda^*$ is also $-1$. Finally, we use pseudoexponential dichotomies to show that in a space of exponentially decreasing functions, the essential spectrum is contained in Re$\lambda \le -\delta <0$.
Citation: Stephen Schecter, Monique Richardson Taylor. Dafermos regularization of a diffusive-dispersive equation with cubic flux. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4069-4110. doi: 10.3934/dcds.2012.32.4069
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