A priori estimates of solutions to local and nonlocal superlinear parabolic problems

TL;DR

This paper develops a priori estimates for solutions to superlinear parabolic problems, including local and nonlocal cases, with applications to blow-up behavior, steady states, and solution properties.

Contribution

It introduces new a priori estimates for sign-changing solutions to superlinear parabolic problems, covering both local and nonlocal operators like the fractional Laplacian.

Findings

Estimates for blow-up rates and energy blow-up

Continuity of blow-up time established

Existence results for nontrivial steady states

Abstract

We consider a priori estimates of possibly sign-changing solutions to superlinear parabolic problems and their applications (blow-up rates, energy blow-up, continuity of blow-up time, existence of nontrivial steady states etc). Our estimates are based mainly on energy, interpolation and bootstrap arguments, but we also use the Pohozaev identity, for example. We first discuss some known results on local problems and then consider problems with nonlocal nonlinearities or nonlocal differential operators. In particular, we deal with the fractional Laplacian and nonlinearities of Choquard type.