An extended variational theory for nonlinear evolution equations via modular spaces

TL;DR

This paper extends the variational theory for nonlinear evolution equations to include non-reflexive and non-separable spaces using modular spaces, enabling broader applicability and well-posedness results.

Contribution

It introduces a novel variational structure based on modular spaces, allowing analysis of evolution equations without reflexivity or polynomial growth restrictions.

Findings

Established a new variational framework for evolution equations in modular spaces.

Proved well-posedness without reflexivity or polynomial growth assumptions.

Applied the theory to equations in Musielak-Orlicz-Sobolev spaces, including variable exponent and double-phase spaces.

Abstract

We propose an extension of the classical variational theory of evolution equations that accounts for dynamics also in possibly non-reflexive and non-separable spaces. The pivoting point is to establish a novel variational structure, based on abstract modular spaces associated to a given convex function. Firstly, we show that the new variational triple is suited for framing the evolution, in the sense that a novel duality paring can be introduced and a generalised computational chain rule holds. Secondly, we prove well-posedness in an extended variational sense for evolution equations, without relying on any reflexivity assumption and any polynomial requirement on the nonlinearity. Finally, we discuss several important applications that can be addressed in this framework: these cover, but are not limited to, equations in Musielak-Orlicz-Sobolev spaces, such as variable exponent, Orlicz, weighted Lebesgue, and double-phase spaces.