Regional, single point, and global blow-up for the fourth-order porous medium type equation with source

TL;DR

This paper investigates different blow-up behaviors in a fourth-order porous medium type equation with a source term, using analytic and numerical methods to analyze critical parameter values and solution structures.

Contribution

It introduces a variational framework for analyzing blow-up solutions at critical parameters, extending solutions via homotopy methods, which is a novel approach in this context.

Findings

Identification of three blow-up types for the equation.

Existence of a countable set of solutions at critical parameters.

Extension of solutions to neighboring parameter values using homotopy.

Abstract

Three types of blow-up for a fourth-order degenerate reaction-diffusion equation are studied by a combination of analytic and numerical methods. At the critical values of parameters, there occurs a variational problem with a countable set of solutions obtained by Lysternik--Shnirelman category theory, which then are extended to neighbouring values of parameters by a homotopy-like approach.