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

This paper is concerned with a class of doubly nonlinear parabolic evolution equations settled in a smooth bounded domain of the Euclidean space and complemented with the initial conditions and with (for simplicity) homogeneous Dirichlet boundary conditions. The diffusion (elliptic) term in the equation is represented by a possibly nonlinear operator, which may range in a very wide class, including the Laplacian, the m-Laplacian for suitable m strictly between 1 and infinity, the “variable-exponent” m-Laplacian, or even some fractional order operators. A second nonlinearity acts on the time derivative of the solution and is assumed to be given by a general class of maximal monotone operators. The main results are devoted to proving existence of weak solutions to the resulting problem, extending the setting considered in previous results related to the variable exponent case. To this end, a theory of subdifferential operators will be established in Musielak–Orlicz spaces satisfying structure conditions of the so-called Delta2-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 to which the abstract result can be applied.