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

Goro Akagi and Giulio Schimperna

Mathematische Nachrichten, 2024, vol. 297, issue 7, 2686-2729

Abstract: This paper is concerned with a parabolic evolution equation of the form A(ut)+B(u)=f$A(u_t) + B(u) = f$, settled in a smooth bounded domain of Rd$\mathbb {R}^d$, d≥1$d\ge 1$, and complemented with the initial conditions and with (for simplicity) homogeneous Dirichlet boundary conditions. Here, −B$-B$ stands for a diffusion operator, possibly nonlinear, which may range in a very wide class, including the Laplacian, the m$m$‐Laplacian for suitable m∈(1,∞)$m\in (1,\infty)$, the “variable‐exponent” m(x)$m(x)$‐Laplacian, or even some fractional order operators. The operator A$A$ is assumed to be in the form [A(v)](x,t)=α(x,v(x,t))$[A(v)](x,t)=\alpha (x,v(x,t))$ with α$\alpha$ being measurable in x$x$ and maximal monotone in v$v$. The main results are devoted to proving existence of weak solutions for a wide class of functions α$\alpha$ that extends the setting considered in previous results related to the variable exponent case where α(x,v)=|v(x)|p(x)−2v(x)$\alpha (x,v)=|v(x)|^{p(x)-2}v(x)$. To this end, a theory of subdifferential operators will be established in Musielak–Orlicz spaces satisfying structure conditions of the so‐called Δ2$\Delta _2$‐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$A$, B$B$) to which the result can be applied.

Date: 2024
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