Traveling wave solutions in a half-space for boundary reactions

TL;DR

This paper establishes the existence, uniqueness, and detailed behavior of traveling wave solutions for a boundary reaction problem in a half-space, highlighting differences from classical interior reaction models.

Contribution

It introduces a variational approach to prove existence and uses comparison principles to establish uniqueness and monotonicity of traveling fronts with explicit decay behaviors.

Findings

Existence of traveling fronts in a half-space with boundary reactions.

Uniqueness of the wave speed and front profile.

Explicit decay rates at infinity for the fronts.

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 non-balanced 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.