On loglog blow-up in higher-order reaction-diffusion and wave equations: A formal “geometric” approach

In 1934, Petrovskii's boundary regularity study of the heat equation gave the first appearance of the ln|ln(T−t)| blow-up factor in PDE theory. We discuss the origin of analogous non-self-similar blow-up in higher-order reaction-diffusion (parabolic) or wave (hyperbolic) equations of the form ut=u2(−uxxxx+u)orutt=u2(−uxxxx+u)in(−L,L)×(0,T), with zero Dirichlet boundary conditions at x=±L, where L>L0∈(π2,π). We present formal arguments that the standard similarity blow-up rate 1T−t acquires an extra universal ln|ln(T−t)| factor. The explanation is based on a “geometric” matching with the so-called logarithmic travelling waves as group invariant solutions of the PDEs. We also discuss connections with log-log blow-up factors occurring in earlier studies of plasma physics parabolic equations and the nonlinear critical Schrödinger equation.