Rothe Time-Semidiscretization for Doubly Nonlinear ParabolicProblems in Musielak–Orlicz–Sobolev Spaces

Abstract

This paper establishes the existence of an entropy solution for a doubly nonlinear parabolic problem set within the framework of Musielak-Orlicz-Sobolev spaces, without imposing the $\Delta_2$ condition. The problem involves a Leray-Lions operator and a general Lipschitz, strictly increasing nonlinearity in the time derivative term, with $L^1$ source data. Our approach employs Rothe's time-semidiscretization method, reducing the evolution problem to a sequence of elliptic entropy subproblems at discrete time steps. We derive uniform a priori estimates in the modular topology associated with the Musielak-Orlicz function, which remain valid in the absence of the $\Delta_2$ assumption. Using these estimates, we prove compactness for the Rothe sequence in $W^{1,x}_0 L_\Psi(Q_T)$ and in $C([0,T]; L^1(\Omega))$. The limit is then identified via monotonicity techniques, confirming that it satisfies the entropy formulation of the original problem. This work unifies and extends previous existence results from standard Sobolev, variable-exponent, and Orlicz-Sobolev settings to the fully Musielak-Orlicz case with general nonlinearities and low-regularity data.