Non self-similar blow-up solutions to the heat equation with nonlinear boundary conditions

TL;DR

This paper constructs finite-time blow-up solutions for the heat equation with nonlinear boundary conditions, demonstrating a blow-up rate different from the self-similar rate in the supercritical case.

Contribution

It introduces new blow-up solutions with non self-similar rates for the heat equation in the supercritical regime.

Findings

Constructed blow-up solutions with non self-similar rates

Demonstrated difference from self-similar blow-up rates in supercritical case

Extended understanding of blow-up behaviors in nonlinear heat equations

Abstract

This paper is concerned with finite blow-up solutions of the heat equation with nonlinear boundary conditions. It is known that a rate of blow-up solutions is the same as the self-similar rate for a Sobolev subcritical case. A goal of this paper is to construct a blow-up solution whose blow-up rate is different from the self-similar rate for a Sobolev supercritical case.