# American Institute of Mathematical Sciences

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May  2017, 37(5): 2619-2651. doi: 10.3934/dcds.2017112

## Traveling fronts bifurcating from stable layers in the presence of conservation laws

 1 Miami University, Department of Mathematics, 301 S. Patterson Ave. Oxford, OH 45056, USA 2 University of Minnesota, School of Mathematics 206 Church St. S.E. Minneapolis, MN 55455, USA

* Corresponding author: Alin Pogan

Received  December 2015 Revised  December 2016 Published  February 2017

Fund Project: AS acknowledges support under grants NSF DMS-1311740 and DMS-1612441. AP acknowledges support by through a Summer Research Grant by College of Arts and Science, Miami University.

We study traveling waves bifurcating from stable standing layers in systems where a reaction-diffusion equation couples to a scalar conservation law. We prove the existence of weekly decaying traveling fronts that emerge in the presence of a weakly stable direction on a center manifold. Moreover, we show the existence of bifurcating traveling waves of constant mass. The main difficulty is to prove the smoothness of the ansatz in exponentially weighted spaces required to apply the Lyapunov-Schmidt methods.

Citation: Alin Pogan, Arnd Scheel. Traveling fronts bifurcating from stable layers in the presence of conservation laws. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2619-2651. doi: 10.3934/dcds.2017112
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