Liouville theorems and universal estimates for superlinear parabolic problems without scale invariance

Abstract

We establish Liouville type theorems in the whole space and in a half-space for parabolic problems without scale invariance. To this end, we employ two methods, respectively based on the corresponding elliptic Liouville type theorems and energy estimates for suitably rescaled problems, and on reduction to a scalar equation by proportionality of components.

We then give applications of known and new Liouville type theorems to universal singularity and decay estimates for non scale invariant parabolic equations and systems involving superlinear nonlinearities with regular variation. To this end, we adapt methods from [33] to parabolic problems.