Entropy solutions for parabolic problems in Musielak framework without using the sign condition of the nonlinearities

TL;DR

This paper establishes the existence of entropy solutions for nonlinear parabolic problems within inhomogeneous Musielak-Orlicz Sobolev spaces without requiring the sign condition on nonlinearities, broadening the scope of solvable problems.

Contribution

It introduces a novel approach to prove existence of solutions without sign conditions and without assuming $ abla_2$ or $ abla_2$ conditions on the Musielak function.

Findings

Existence of entropy solutions in Musielak-Orlicz spaces without sign conditions.

Solutions accommodate data in $L^1(Q)$.

No $ abla_2$ or $ abla_2$ assumptions on the Musielak function.

Abstract

We prove in this paper the existence of solutions of nonlinear parabolic problems in inhomogeneous Musielak Orlicz Sobolev spaces, we assume neither a $\Delta_2$ nor $\nabla_2$ on the Musielak function $\varphi$. The main contribution of our work is to prove the existence of entropy solutions without the sign condition on the nonlinearity. The second term $f$ belongs to $L^1(Q)$.