On root log log blow-up in higher-order reaction-diffusion and wave equations: A formal 'geometric' approach

摘要

In 1934, Petrovskii's boundary regularity study of the heat equation gave the first appearance of the root In vertical bar In(T - t)vertical bar 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 u(t) = u(2)(-u(xxxx) + u) or u(tt) = u(2)(-u(xxxx) + u) in (-L. L) x (0, T), with zero Dirichlet boundary conditions at x = +/- L, where L > L-0 is an element of (pi/2, pi). We present formal arguments that the standard similarity blow-up rate 1/root T-t acquires an extra universal root In vertical bar In(T - t)vertical bar 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 Schrodinger equation.