Five types of blow-up in a semilinear fourth-order reaction-diffusion equation: an analytic-numerical approach

TL;DR

This paper classifies five distinct blow-up patterns in a fourth-order reaction-diffusion equation using combined analytical and numerical methods, addressing a less-explored area with implications for higher-order diffusion models.

Contribution

It introduces a comprehensive classification of blow-up behaviors for a fourth-order semilinear PDE, expanding understanding beyond the well-studied second-order cases.

Findings

Identifies five blow-up types in the fourth-order equation

Provides analytical descriptions of each blow-up pattern

Uses numerical simulations to validate theoretical results

Abstract

Five types of blow-up patterns that can occur for the 4th-order semilinear parabolic equation of reaction-diffusion type $$ u_t= -\Delta^2 u + |u|^{p-1} u \quad {in} \quad \ren \times (0,T), p>1, \quad \lim_{t \to T^-}\sup_{x \in \ren} |u(x,t)|= +\iy, $$ are discussed. For the semilinear heat equation $u_t= \Delta u+ u^p$, various blow-up patterns were under scrutiny since 1980s, while the case of higher-order diffusion was studied much less, regardless a wide range of its application.