Subdifferential calculus and doubly nonlinear evolution equations in L^p spaces with variable exponents

TL;DR

This paper investigates the existence and regularity of solutions for a class of doubly nonlinear parabolic equations with variable exponents, introducing new calculus tools and analyzing functional space relations.

Contribution

It develops a chain rule for convex functionals in variable exponent spaces and clarifies measurability and space relations in this complex setting.

Findings

Proved existence of strong solutions.

Established regularity results.

Formulated a chain rule for nonsmooth convex functionals.

Abstract

This paper is concerned with the Cauchy-Dirichlet problem for a doubly nonlinear parabolic equation involving variable exponents and provides some theorems on existence and regularity of strong solutions. In the proof of these results, we also analyze the relations occurring between Lebesgue spaces of space-time variables and Lebesgue-Bochner spaces of vector-valued functions, with a special emphasis on measurability issues and particularly referring to the case of space-dependent variable exponents. Moreover, we establish a chain rule for (possibly nonsmooth) convex functionals defined on variable exponent spaces. Actually, in such a peculiar functional setting the proof of this integration formula is nontrivial and requires a proper reformulation of some basic concepts of convex analysis, like those of resolvent, of Yosida approximation, and of Moreau-Yosida regularization.