Nonlinear parabolic problems in Musielak--Orlicz spaces

TL;DR

This paper investigates the existence of weak solutions for nonlinear parabolic problems within Musielak-Orlicz spaces, utilizing maximal monotone graphs and density arguments under specific regularity conditions.

Contribution

It introduces a framework for analyzing parabolic problems with multi-valued terms in Musielak-Orlicz spaces, extending previous methods to more general growth and coercivity conditions.

Findings

Established existence of weak solutions under new conditions

Developed uniform boundedness results for convolution operators

Applied logarithmic Hölder regularity assumptions

Abstract

Our studies are directed to the existence of weak solutions to a parabolic problem containing a multi-valued term. The problem is formulated in the language of maximal monotone graphs. We assume that the growth and coercivity conditions of a nonlinear term are prescribed by means of time and space dependent $N$--function. This results in formulation of the problem in generalized Musielak-Orlicz spaces. We are using density arguments, hence an important step of the proof is a uniform boundedness of appropriate convolution operators in Musielak-Orlicz spaces. For this purpose we shall need to assume a kind of logarithmic H\"older regularity with respect to $t$ and $x$.