The evolution to localized and front solutions in a non-Lipschitz reaction-diffusion Cauchy problem with trivial initial data

TL;DR

This paper proves the existence of localized and front solutions in a non-Lipschitz reaction-diffusion equation with trivial initial data, using dynamical systems methods to analyze self-similar solutions.

Contribution

It establishes the existence of classical self-similar solutions and heteroclinic/homoclinic connections in a non-Lipschitz reaction-diffusion problem with trivial initial data.

Findings

Existence of a two-parameter family of homoclinic connections.

Existence of heteroclinic connections between equilibrium points.

Bounds on the convergence rate of solutions.

Abstract

In this paper, we establish the existence of spatially inhomogeneous classical self-similar solutions to a non-Lipschitz semi-linear parabolic Cauchy problem with trivial initial data. Specifically we consider bounded solutions to an associated two-dimensional non-Lipschitz non-autonomous dynamical system, for which, we establish the existence of a two-parameter family of homoclinic connections on the origin, and a heteroclinic connection between two equilibrium points. Additionally, we obtain bounds and estimates on the rate of convergence of the homoclinic connections to the origin.