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

Alin Pogan; Arnd Scheel

doi:10.3934/dcds.2017112

Article Contents

2017, Volume 37, Issue 5: 2619-2651. Doi: 10.3934/dcds.2017112

This issue Previous Article Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential Next Article Existence and uniqueness of positive solutions for a class of logistic type elliptic equations in $\mathbb{R}^N$ involving fractional Laplacian

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

Alin Pogan^1, , and
Arnd Scheel^2,

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
* Corresponding author: Alin Pogan

Received: December 2015

Revised: December 2016

Published online: February 2017

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.

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Abstract

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.

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Mathematics Subject Classification: Primary: 37L15.

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