Incomplete self-similar blow-up in a semilinear fourth-order reaction-diffusion equation

TL;DR

This paper demonstrates that solutions to a fourth-order reaction-diffusion equation can be extended beyond blow-up time through self-similar solutions, revealing incomplete self-similar blow-up and discussing other blow-up types.

Contribution

It introduces the concept of incomplete self-similar blow-up and explores the possibility of extending solutions beyond blow-up in a fourth-order reaction-diffusion context.

Findings

Self-similar blow-up is generally incomplete for the studied equation.

Solutions can be extended after blow-up via self-similar extensions.

Other non self-similar blow-up types are also discussed.

Abstract

It is shown that self-similar blow-up for a fourth-order reaction-diffusion equation is incomplete in the sense that, in general, there exists a self-similar extension of solutions after blow-up. Other types of complete blow-up of non self-similar form are discussed.