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 $A(u_t) + B(u) = f$, settled in a smooth bounded domain of ${\bf R}^d$, $d \geq 1$, and complemented with the initial conditions and with (for simplicity) homogeneous Dirichlet boundary conditions. Here, $-B$ stands for a diffusion operator, possibly nonlinear, which may range in a very wide class, including the Laplacian, the $m$-Laplacian for suitable $m\in(1,\infty)$), the "variable-exponent" $m(x)$-Laplacian, or even some fractional order operators. The operator $A$ is assumed to be in the form $[A(v)](x, t) = \alpha(x, v(x, t))$ with $\alpha$ being measurable in $x$ and maximal monotone in $v$. The main results are devoted to proving existence of weak solutions for a wide class of functions $\alpha$ that extends the setting considered in previous results related to the variable exponent case where $\alpha(x, v) = |v(x)|^{p(x)-2} v(x)$. To this end, a theory of subdifferential operators will be established in Musielak-Orlicz spaces satisfying structure conditions of the so-called $\Delta_2$-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 $A$, $B$) to which the result can be applied.