A non-smooth regularization of a forward-backward parabolic equation

TL;DR

This paper introduces a regularized model for diffusion described by a forward-backward parabolic equation, establishing existence, uniqueness, and continuous dependence of solutions for a system relevant to physical quantities.

Contribution

It presents a novel regularization approach for a non-convex diffusion model using a maximal monotone graph, ensuring well-posedness of the system.

Findings

Proved existence and uniqueness of solutions.

Established continuous dependence on initial data.

Developed a viscous regularization method for non-convex free-energy densities.

Abstract

In this paper we introduce a model describing diffusion of species by a suitable regularization of a "forward-backward" parabolic equation. In particular, we prove existence and uniqueness of solutions, as well as continuous dependence on data, for a system of partial differential equations and inclusion, which may be interpreted, e.g., as evolving equation for physical quantities such as concentration and chemical potential. The model deals with a constant mobility and it is recovered from a possibly non-convex free-energy density. In particular, we render a general viscous regularization via a maximal monotone graph acting on the time derivative of the concentration and presenting a strong coerciveness property.