Standing Sphere Blow-up Solutions to The Nonlinear Heat Equation

TL;DR

This paper constructs and proves the existence and stability of finite-time blow-up ring solutions for the nonlinear heat equation in the radial case, confirming previous predictions.

Contribution

It provides a rigorous proof of ring blow-up solutions and their stability, advancing understanding of singularity formation in nonlinear heat equations.

Findings

Existence of finite-time ring blow-up solutions.

Stability of these solutions among radially symmetric solutions.

Confirmation of previous formal predictions.

Abstract

In this paper, we construct a singular standing ring solution of the nonlinear heat in the radial case. We give rigorous proof for the existence of a ring blow-up solution in finite time. This result was predicted formally by Baruch, Fibich and Gavish \cite{BFGpd10}. We also prove the stability of these dynamics among radially symmetric solutions.