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

TL;DR

This paper proves Liouville theorems for superlinear parabolic equations lacking scale invariance, providing universal estimates for singularity and decay, and introduces new methods for such non-scale-invariant problems.

Contribution

It develops novel Liouville theorems for non-scale-invariant parabolic problems and applies these to derive universal estimates, expanding the understanding of superlinear parabolic equations.

Findings

Liouville theorems established for parabolic problems without scale invariance

Universal singularity and decay estimates derived for such equations

Methods adapted from previous work to handle non-scale-invariant cases

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 our preprint arXiv:2407.04154 to parabolic problems.