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

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.