Stochastic parabolic equations in Musielak-Orlicz spaces with discontinuous in time N-function

TL;DR

This paper establishes the existence of weak solutions for stochastic parabolic equations with growth conditions governed by a time-discontinuous Musielak-Orlicz N-function, extending the applicability to complex models like p(t,x)-Laplacian.

Contribution

It introduces a framework for analyzing stochastic parabolic PDEs with non-regular N-functions, including a proof of Itô's formula in Orlicz spaces, broadening the scope of existing models.

Findings

Existence of weak solutions under general N-function conditions

Extension of Itô's formula to Musielak-Orlicz spaces

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

Abstract

We consider a stochastic parabolic partial differential equation with Dirichlet boundary conditions, multiplicative stochastic noise, and a monotone parabolic operator A. The growth and coercivity of A is controlled by a general N-function M, which depends on time, and spatial variable, but we do not assume any regularity with respect to the former. We show the existence of weak solutions to such system. As auxiliary result, we also provide the proof for the It\^{o}'s formula in Orlicz spaces. This general result applies to the ones studied in the literature, such as p(t, x)-Laplacian and double phase problems.