Universal estimates and Liouville theorems for superlinear problems without scale invariance

TL;DR

This paper extends rescaling methods to nonlinear elliptic and parabolic problems lacking scale invariance, establishing new Liouville theorems and universal estimates for a broader class of nonlinearities.

Contribution

It introduces modified rescaling techniques applicable to non scale invariant nonlinearities, linking universal estimates with Liouville theorems in this expanded setting.

Findings

Liouville theorems for non scale invariant nonlinearities in parabolic problems

Universal space-time estimates for solutions without scale invariance

New bounds on blow-up rates and decay in elliptic and parabolic equations

Abstract

We revisit rescaling methods for nonlinear elliptic and parabolic problems and show that, by suitable modifications, they may be used for nonlinearities that are not scale invariant even asymptotically and whose behavior can be quite far from power like. In this enlarged framework, by adapting the doubling-rescaling method from [37, 38], we show that the equivalence found there between universal estimates and Liouville theorems remains valid. In the parabolic case we also prove a Liouville type theorem for a rather large class of non scale invariant nonlinearities. This leads to a number of new results for non scale invariant elliptic and parabolic problems, concerning space or space-time singularity estimates, initial and final blow-up rates, universal and a priori bounds for global solutions, and decay rates in space and/or time. We illustrate our approach by a number of examples, which in turn give indication about the optimality of the estimates and of the assumptions.