Parabolic equations in Musielak -- Orlicz spaces with discontinuous in time $N$-function

TL;DR

This paper establishes the existence and uniqueness of weak solutions for a class of parabolic PDEs with non-standard growth conditions in Musielak-Orlicz spaces, even when the defining $N$-function is discontinuous in time.

Contribution

It introduces a novel approach to handle discontinuous time-dependent $N$-functions in parabolic equations, extending existing results to more general growth conditions.

Findings

Existence of weak solutions under minimal regularity assumptions.

Uniqueness of solutions in the isotropic case.

Applicability to $p(t,x)$-Laplacian and double-phase problems.

Abstract

We consider a parabolic PDE with Dirichlet boundary condition and monotone operator $A$ with non-standard growth controlled by an $N$-function depending on time and spatial variable. We do not assume continuity in time for the $N$-function. Using an additional regularization effect coming from the equation, we establish the existence of weak solutions and in the particular case of isotropic $N$-function, we also prove their uniqueness. This general result applies to equations studied in the literature like $p(t,x)$-Laplacian and double-phase problems.