Blow-up in higher-order reaction-diffusion and wave equations: how √(log log) factor occurs

2009/01/27 by V. A. Galaktionov, Galaktionov, V. A.
Mathematics · #35K40 #35K55 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35K40 #msc:35K55

paper · pdf · doi:10.48550/arxiv.0901.4314

41 pages, 10 figures

arxiv created 2009/01/27 · arxiv updated 2009/12/01

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 Schrödinger 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.

Related

• Blowups for a class of second order nonlinear hyperbolic equations: A reduced model of nonlinear Jeans instability
• A log-log upper bound on blow-up rates for the mass-critical half-wave equation
• Towards the KPP--Problem and log t-Front Shift for Higher-Order Nonlinear PDEs III. Dispersion and Hyperbolic Equations
• Non self-similar blow-up solutions to the heat equation with nonlinear boundary conditions
• Incomplete self-similar blow-up in a semilinear fourth-order reaction-diffusion equation