Some local properties of subsolution and supersolutions for a doubly nonlinear nonlocal p-Laplace equation

Abstract

We establish a local boundedness estimate for weak subsolutions to a doubly nonlinear parabolic fractional p-Laplace equation. Our argument relies on energy estimates and a parabolic nonlocal version of De Giorgi's method. Furthermore, by means of a new algebraic inequality, we show that positive weak supersolutions satisfy a reverse Hölder inequality. Finally, we also prove a logarithmic decay estimate for positive supersolutions.

Large time behavior for a nonlocal nonlinear gradient flow

“…is proved to be exponential. We also seize the opportunity to mention the recent papers [BGK20] and [BGK21], where another nonlocal doubly nonlinear equation is studied.…”

Local boundedness of a mixed local–nonlocal doubly nonlinear equation

“…Very recently, Brasco, Lindgren and Strömqvist [12] showed the Hölder regularity for (local) weak solutions in the case p ≧ 2, whose proof is completely different to previous approaches in view of using the iterated discrete differentiation method together with a Morrey type embedding. On the contrast, in the doubly nonlinear setting, Banerjee, Garain and Kinnunen [3] first proved with the local boundedness of positive solutions to (1.5). The regular form of (1.1), that is,…”

Hölder regularity for parabolic fractional $p$-Laplacian

“…The parabolic nonlocal problem (1.1) has witnessed a growing interest recently; see [2,3,4,6,15,16,18,19,21,23,24,26,27] -just to mention a few. Coming to the local regularity, while the case p = 2 has been subject to extensive studies, the case p = 2 is largely open.…”

“…Coming to the local regularity, while the case p = 2 has been subject to extensive studies, the case p = 2 is largely open. Local regularity of supersolutions is studied in [2,3]. Some local boundedness estimates are reported in [15,23].…”