Parabolic equations with non-standard growth and measure or integrable data

TL;DR

This paper proves the existence and uniqueness of weak and entropy solutions for a class of parabolic equations with non-standard growth conditions and measure or integrable data, under specific regularity assumptions on the variable exponent.

Contribution

It introduces new existence and uniqueness results for parabolic equations with non-standard growth and variable exponent, relaxing smoothness assumptions on the time variable.

Findings

Existence of weak and entropy solutions under given conditions.

Uniqueness of entropy solutions with boundedness and log-Hölder continuity of the exponent.

Solutions accommodate measure or L^1 data in the equation.

Abstract

We consider a parabolic partial differential equation with Dirichlet boundary conditions and measure or $L^1$ data. The key difficulty consists in a presence of a monotone operator~$A$ subjected to a non-standard growth condition, controlled by the exponent $p$ depending on the time and the spatial variable. We show the existence of a weak and an entropy solution to our system, as well as the uniqueness of an entropy solution, under the assumption of boundedness and log-H\"{o}lder continuity of the variable exponent~$p$ with respect to the spatial variable. On the other hand, we do not assume any smoothness of~$p$ with respect to the time variable.