Global solution and long-time behavior for a problem of phase segregation of the Allen-Cahn type

TL;DR

This paper analyzes a phase segregation model combining PDE and ODE components, proving global existence, uniqueness, and characterizing long-term behavior of solutions, with implications for Allen-Cahn type equations with memory effects.

Contribution

It introduces a novel reduction of a coupled PDE-ODE system to an Allen-Cahn equation with memory, establishing global solutions and their asymptotic properties.

Findings

Proved global existence and uniqueness of smooth solutions.

Characterized the omega-limit set of solutions.

Reduced the coupled system to an Allen-Cahn equation with memory.

Abstract

In this paper we study a model for phase segregation consisting in a sistem of a partial and an ordinary differential equation. By a careful definition of maximal solution to the latter equation, this system reduces to an Allen-Cahn equation with a memory term. Global existence and uniqueness of a smooth solution are proven and a characterization of the omega-limit set is given.