Global attractors for doubly nonlinear evolution equations with non-monotone perturbations

TL;DR

This paper develops an abstract framework for analyzing global attractors in doubly nonlinear evolution equations with non-monotone perturbations, using generalized semiflows to handle non-uniqueness of solutions.

Contribution

It introduces a novel approach based on generalized semiflows for constructing global attractors in non-monotone doubly nonlinear equations, extending existing theories.

Findings

Established existence of global attractors for the abstract class of equations.

Applied the theory to a generalized Allen-Cahn equation with non-convex potential.

Analyzed a semilinear parabolic equation with gradient-dependent nonlinearities.

Abstract

This paper proposes an abstract theory concerned with dynamical systems generated by doubly nonlinear evolution equations governed by subdifferential operators with non-monotone perturbations in a reflexive Banach space setting. In order to construct global attractors, an approach based on the notion of generalized semiflow is employed instead of the usual semi-group approach, since solutions of the Cauchy problem for the equation might not be unique. Moreover, the preceding abstract theory is applied to a generalized Allen-Cahn equation whose potential is divided into a convex part and a non-convex part as well as a semilinear parabolic equation with a nonlinear term involving gradients.