On 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 u t = u 2 ( − u x x x x + u ) or u t t = u 2 ( − u x x x x + u ) in ( − L , L ) × ( 0 , T ) , with zero Dirichlet boundary conditions at x = ± L , where L > L 0 ∈ ( π 2 , π ) . We present formal arguments that the standard similarity blow-up rate 1 T − 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.