On a spectrum of blow-up patterns for a higher-order semilinear parabolic equation

We describe a spectrum of blow-up patterns for the 2mth-order semilinear parabolic equation, u(t) - 1(-Delta)(m)u + \u \ (p), x is an element ofR(N), t > 0; m >1, p >1. This problem is well understood in the second-order case m = 1 (the semilinear heat equation), where for p less than or equal to p(s) = (N + 2)/(N - 2)+ the inner space-time structure of blow-up patterns is locally governed by separable Hermite polynomials of arbitrary finite order as eigenfunctions of a linear self-adjoint differential operator. We consider m > 1 and describe a principle idea of constructing of a spectrum of blow-up patterns which in the inner self-similar region are composed of a countable subset of separable Kummer's polynomials, which are eigenfunctions of a non-self-adjoint linear differential operator. A matching procedure extends such linearized structures to the nonlinear intermediate region where the asymptotic blow-up behaviour is described by a unique self-similar solution of the first-order Hamilton-Jacobi equation.