Existence of peaking solutions for semilinear heat equations with blow-up profile above the singular steady state

Abstract

We consider positive solutions of the semilinear heat equation with supercritical power nonlinearity, and construct peaking solutions by connecting a backward self-similar solution with a forward self-similar solution. In particular, we show the existence of incomplete blow-up solutions with blow-up profile above the singular steady state. We also show the existence of a minimal continuation and obtain some properties of non-minimal continuations.