On two-signed solutions to a second order semi-linear parabolic partial differential equation with non-Lipschitz nonlinearity

TL;DR

This paper proves the existence of a family of radially symmetric self-similar solutions to a semi-linear parabolic PDE with non-Lipschitz nonlinearity, analyzing their behavior and convergence properties.

Contribution

It establishes the existence of a parameterized family of solutions and analyzes their asymptotic behavior for a challenging non-Lipschitz PDE.

Findings

Existence of a 1-parameter family of solutions

Solutions converge algebraically to the origin

Solutions oscillate as the spatial variable tends to infinity

Abstract

In this paper, we establish the existence of a 1-parameter family of spatially inhomogeneous radially symmetric classical self-similar solutions to a Cauchy problem for a semi-linear parabolic PDE with non-Lipschitz nonlinearity and trivial initial data. Specifically we establish well-posedness for an associated initial value problem for a singular two-dimensional non-autonomous dynamical system with non-Lipschitz nonlinearity. Additionally, we establish that solutions to the initial value problem converge algebraically to the origin and oscillate as $\eta\to \infty$.