A priori bounds for degenerate and singular evolutionary partial integro-differential equations

TL;DR

This paper establishes global boundedness and a maximum principle for a class of degenerate and singular quasilinear evolutionary partial integro-differential equations, including time fractional p-Laplace equations, using energy estimates and De Giorgi's iteration.

Contribution

It provides new a priori bounds and maximum principles for complex integro-differential equations with degeneracy and singularity, extending existing PDE theory.

Findings

Global boundedness of weak solutions

Validity of maximum principle for these equations

Extension to time fractional p-Laplace equations

Abstract

We study quasilinear evolutionary partial integro-differential equations of second order which include time fractional $p$-Laplace equations of time order less than one. By means of suitable energy estimates and De Giorgi's iteration technique we establish results asserting the global boundedness of appropriately defined weak solutions of these problems. We also show that a maximum principle is valid for such equations.