Non-uniqueness and global similarity solutions for a higher-order semilinear parabolic equation

Abstract

We consider a 2mth-order (m ≥ 2) semilinear parabolic equation of the reaction–diffusion type, and initial data , q ≥ 1. This is a higher-order extension of the classical semilinear heat equation for m = 1 from combustion theory. It is well known from Weissler's results that, for p < p0 = 1 + 2mq/N, there exists a unique local in time solution as a continuous curve for sufficiently small T > 0. For m = 1, it was proved that local nonexistence can happen for p > p0. In the case m = 1, which was studied in greater detail in the 1980s, the precise range p ≤ 1 + 2q/N for uniqueness of such solutions has been established. For p > 1 + 2q/N, non-uniqueness was proved by Haraux and Weissler by constructing special similarity solutions.Our goal is to show that non-uniqueness takes place for the higher-order parabolic equations if p > p0. To this end, we describe a discrete subset of similarity solutions where each V is a radial, exponentially decaying solution of the elliptic equation By perturbation techniques, we establish the existence of radially symmetric similarity profiles Vl for p close to critical bifurcation exponents pl = 1 + 2m/(N + l), l = 0, 2, …, and prove that all the p-bifurcation branches remain in the subcritical Sobolev range p < pS = (N + 2m)/(N − 2m)+. By using analytic, asymptotic and numerical methods we justify some global properties of the bifurcation diagram. We also demonstrate that the similarity profiles satisfy Sturm's zero property in a certain ‘approximate’ sense.