Blow-Up Profiles and Dynamics in Negative Time for the Semilinear Heat Equation

Abstract

We investigate the blow-up behavior of solutions to the semilinear heat equation ut=uxx+|u|p−1u,x∈R,u∈R, for exponents 1<p<1+2m, where m∈N denotes the number of positive eigenvalues of the linearized operator in similarity variables, equivalently the dimension of the associated unstable manifold, which determines both the admissible exponent range and the structure of the blow-up profiles. We construct solutions that exist on the interval −1<t<T and become unbounded both as t→−1 (backward blow-up) and as t→T (forward blow-up). At blow-up time, the solution profile exhibits a finite number of critical values, which can be prescribed in advance, and possesses a structure with m+1 monotonicity intervals. By introducing similarity variables, we reduce the problem to an evolution equation in weighted spaces and identify the role of unstable manifolds. Our results establish a classification of blow-up dynamics in terms of spectral properties and provide a systematic framework for constructing solutions with prescribed spatial patterns of singularity.