On degenerate blow-up profiles for the subcritical semilinear heat equation

TL;DR

This paper constructs a novel, cross-shaped blow-up profile for solutions of the subcritical semilinear heat equation, demonstrating the existence of diverse blow-up behaviors localized at a single point.

Contribution

It introduces a new type of blow-up profile and a general method for constructing solutions with various localized blow-up behaviors.

Findings

Existence of a cross-shaped blow-up profile at the origin.

Method extends to constructing multiple blow-up solutions with different profiles.

Demonstrates the diversity of blow-up behaviors in subcritical heat equations.

Abstract

We consider the semilinear heat equation with a superlinear power nonlinearity in the Sobolev subcritical range. We construct a solution which blows up in finite time only at the origin, with a completely new blow-up profile, which is cross-shaped. Our method is general and extends to the construction of other solutions blowing up only at the origin, with a large variety of blow-up profiles, degenerate or not.