Bistable travelling waves for nonlocal reaction diffusion equations

TL;DR

This paper studies bistable travelling wave solutions in nonlocal reaction-diffusion equations without the comparison principle, constructing solutions connecting 0 to an unknown steady state and analyzing their properties.

Contribution

It introduces a method to construct bistable travelling waves in nonlocal equations without relying on the comparison principle, including the case of focusing kernels.

Findings

Constructed travelling wave solutions connecting 0 to an unknown steady state.

Proved that for focusing kernels, the wave connects 0 to 1.

Results extend to nonlocal ignition cases.

Abstract

We are concerned with travelling wave solutions arising in a reaction diffusion equation with bistable and nonlocal nonlinearity, for which the comparison principle does not hold. Stability of the equilibrium $u\equiv 1$ is not assumed. We construct a travelling wave solution connecting 0 to an unknown steady state, which is "above and away", from the intermediate equilibrium. For focusing kernels we prove that, as expected, the wave connects 0 to 1. Our results also apply readily to the nonlocal ignition case.