The Time Derivative in a Singular Parabolic Equation

TL;DR

This paper proves that weak solutions to the singular evolutionary p-Laplace equation possess a time derivative in Sobolev's sense, which is locally integrable to some power greater than one, advancing understanding of regularity in such equations.

Contribution

It establishes the existence and local integrability of the time derivative for solutions to the singular p-Laplace equation, a novel regularity result for this class of equations.

Findings

Weak solutions have a Sobolev time derivative.

The time derivative is locally summable to a power greater than one.

Advances regularity theory for singular parabolic equations.

Abstract

We study the Evolutionary p-Laplace Equation in the singular case 1 < p < 2. We prove that a weak solution has a time derivative in Sobolev's sense and that the time derivative is locally summable to some power > 1.