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

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.

arXiv comment29 pages, 2 figures

Cited by in corpus (2)

• On two-signed solutions to a second order semi-linear parabolic partial differential equation with non-Lipschitz nonlinearity
• On a $L^{\infty }$ functional derivative estimate relating to the Cauchy problem for scalar semi-linear parabolic partial differential equations with general continuous nonlinearity