Traveling waves and weak solutions for an equation with degenerate dispersion

Ambrose, David; Wright, J.

doi:10.1090/S0002-9939-2013-12070-8

Traveling waves and weak solutions for an equation with degenerate dispersion
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by David M. Ambrose and J. Douglas Wright PDF

Proc. Amer. Math. Soc. 141 (2013), 3825-3838 Request permission

Abstract:

We consider the following family of equations: \begin{equation*} u_t = 2 u u_{xxx} - u_x u_{xx} + 2 k u u_x. \end{equation*} Here $k \ne 0$ is a constant and $x \in [-L_0,L_0]$. We demonstrate that for these equations there are compactly supported traveling wave solutions (which are in $H^2$), and the Cauchy problem (with $H^2$ initial data) possesses a weak solution which exists locally in time. These are the first degenerate dispersive evolution PDE where both of these features are known to hold simultaneously. Moreover, if $k < 0$ or $L_0$ is not too large, the solution exists globally in time.

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