Electron. J. Diff. Equ., Vol. 2013 (2013), No. 39, pp. 1-15.

Existence of traveling waves for diffusive-dispersive conservation laws

Cezar I. Kondo, Alex F. Rossini

Abstract:
In this work we show the existence existence and uniqueness of traveling waves for diffusive-dispersive conservation laws with flux function in $C^{1}(\mathbb{R})$, by using phase plane analysis. Also we estimate the domain of attraction of the equilibrium point attractor corresponding to the right-hand state. The equilibrium point corresponding to the left-hand state is a saddle point. According to the phase portrait close to the saddle point, there are exactly two semi-orbits of the system. We establish that only one semi-orbit come in the domain of attraction and converges to $(u_{-},0)$ as $y\to -\infty$. This provides the desired saddle-attractor connection.

Submitted October 1, 2012. Published February 1, 2013.
Math Subject Classifications: 35L65, 76N10.
Key Words: Scalar conservation law; diffusive-dispersive; weak solution; traveling wave; phase portrait.

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Cezar I. Kondo
Federal University of Sao Carlos
Department of Mathematics
P. O. Box 676 13565-905, Sao Carlos - SP, Brazil
email: dcik@dm.ufscar.br
Alex F. Rossini
Federal University of Sao Carlos
Department of Mathematics
P. O. Box 676 13565-905, Sao Carlos - SP, Brazil
email: alexrossini@dm.ufscar.br

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