Lower semicontinuity and pointwise behavior of supersolutions for some doubly nonlinear nonlocal parabolic $p$-Laplace equations

TL;DR

This paper investigates the pointwise behavior and lower semicontinuity of weak supersolutions for a class of doubly nonlinear nonlocal parabolic p-Laplace equations, extending previous local and elliptic results to the nonlocal parabolic context.

Contribution

It establishes lower semicontinuity and pointwise characterization of supersolutions for nonlocal parabolic equations, including fractional p-Laplace and porous medium equations.

Findings

Supersolutions have lower semicontinuous representatives.

The semicontinuous representative is determined by past values.

Results extend local and elliptic cases to nonlocal parabolic equations.

Abstract

We discuss pointwise behavior of weak supersolutions for a class of doubly nonlinear parabolic fractional $p$-Laplace equations which includes the fractional parabolic $p$-Laplace equation and the fractional porous medium equation. More precisely, we show that weak supersolutions have lower semicontinuous representative. We also prove that the semicontinuous representative at an instant of time is determined by the values at previous times. This gives a pointwise interpretation for a weak supersolution at every point. The corresponding results hold true also for weak subsolutions. Our results extend some recent results in the local parabolic case and in the nonlocal elliptic case to the nonlocal parabolic case. We prove the required energy estimates and measure theoretic De Giorgi type lemmas in the fractional setting.