Flat Standing Sphere Blow-up Solutions for the Nonlinear Heat Equation

TL;DR

This paper constructs a specific singular solution for the nonlinear heat equation that blows up on a sphere, providing detailed asymptotics and refining existing methods for analyzing such phenomena.

Contribution

It introduces a new flat blow-up solution with explicit asymptotics, extending the method of Merle and Zaag to the spherical case.

Findings

Existence of a finite-time singularity on a sphere

Explicit asymptotic description of the blow-up profile

Refinement of the finite-dimensional reduction method

Abstract

In this paper, we prove the existence of a singular standing sphere blow-up solution for the nonlinear heat equation with radial symmetry. This solution develops a finite-time singularity on a fixed-radius sphere and exhibits a flat blow-up profile. Our construction refines the method developed by Merle and Zaag \cite{MZJEMS24} which reduces the infinite-dimensional dynamics to a finite-dimensional problem in radial case. The solution satisfies explicit asymptotics near the singular ring and remains regular elsewhere.