On a class of doubly nonlinear evolution equations in Musielak–Orlicz spaces

Abstract

This paper is concerned with a parabolic evolution equation of the form , settled in a smooth bounded domain of , , and complemented with the initial conditions and with (for simplicity) homogeneous Dirichlet boundary conditions. Here, stands for a diffusion operator, possibly nonlinear, which may range in a very wide class, including the Laplacian, the ‐Laplacian for suitable , the “variable‐exponent” ‐Laplacian, or even some fractional order operators. The operator is assumed to be in the form with being measurable in and maximal monotone in . The main results are devoted to proving existence of weak solutions for a wide class of functions that extends the setting considered in previous results related to the variable exponent case where . To this end, a theory of subdifferential operators will be established in Musielak–Orlicz spaces satisfying structure conditions of the so‐called ‐type, and a framework for approximating maximal monotone operators acting in that class of spaces will also be developed. Such a theory is then applied to provide an existence result for a specific equation, but it may have an independent interest in itself. Finally, the existence result is illustrated by presenting a number of specific equations (and, correspondingly, of operators , ) to which the result can be applied.