Blow-up in higher-order reaction-diffusion and wave equations: how $\sqrt{log log}$ factor occurs

TL;DR

This paper investigates the mechanisms behind non self-similar blow-up phenomena in higher-order reaction-diffusion and wave equations, highlighting the emergence of the log log factor and its connections to classical PDE results.

Contribution

It provides a comprehensive explanation of the origin of the log log blow-up factor in complex PDEs, linking it to historical and physical contexts.

Findings

Identification of the log log blow-up factor in various PDEs

Connections between blow-up behavior and plasma physics and Schrd6dinger equations

Historical link to Petrovskii's boundary regularity results

Abstract

The origin of non self-similar blow-up in higher-order reaction-diffusion (parabolic), wave (hyperbolic) and nonlinear dispersion equations is explained by a combination of various methods. Some links and similarities with double-log blow-up terms occurring in earlier studies of plasma physics second-order parabolic equations and the nonlinear critical Schr\"odinger equation are discussed. On the other hand, the log-log factor obtained in Petrovskii's boundary regularity study of a paraboloid vertex for the heat equation in 1934 was the first its appearance in PDE theory.