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 $A\left(u_t\right)+B\left(u\right)=f$, settled in a smooth bounded domain of $R^d$, $d\ge 1$, 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” $m\left(x\right)$-Laplacian, or even some fractional order operators. The operator $A$ is assumed to be in the form $\left[A\right(v\left)\right](x,t)=\alpha (x,v(x,t\left)\right)$ 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 $\alpha (x,v)={\left|v\left(x\right)\right|}^{p\left(x\right)-2}v\left(x\right)$. 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.