Vol. 8, No. 2, 2015

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Traveling wave solutions in a half-space for boundary reactions

Xavier Cabré, Neus Cónsul and José V. Mandé

Vol. 8 (2015), No. 2, 333–364
Abstract

We prove the existence and uniqueness of a traveling front and of its speed for the homogeneous heat equation in the half-plane with a Neumann boundary reaction term of unbalanced bistable type or of combustion type. We also establish the monotonicity of the front and, in the bistable case, its behavior at infinity. In contrast with the classical bistable interior reaction model, its behavior at the side of the invading state is of power type, while at the side of the invaded state its decay is exponential. These decay results rely on the construction of a family of explicit bistable traveling fronts. Our existence results are obtained via a variational method, while the uniqueness of the speed and of the front rely on a comparison principle and the sliding method.

Keywords
traveling front, traveling wave, boundary reaction, homogeneous heat equation, bistable nonlinearity, combustion nonlinearity
Mathematical Subject Classification 2010
Primary: 35J65, 35K57
Milestones
Received: 23 April 2014
Accepted: 26 November 2014
Published: 10 May 2015
Authors
Xavier Cabré
Icrea and Universitat Politécnica de Catalunya
Departament de Matemática Aplicada I
Diagonal 647
08028 Barcelona
Spain
Neus Cónsul
Universitat Politécnica de Catalunya
Departament de Matemática Aplicada I
Diagonal 647
08028 Barcelona
Spain
José V. Mandé
Universitat Politécnica de Catalunya
Departament de Matemática Aplicada I
Diagonal 647
08028 Barcelona
Spain