Non-isothermal viscous Cahn--Hilliard equation with inertial term and dynamic boundary conditions

TL;DR

This paper extends the analysis of a non-isothermal viscous Cahn--Hilliard equation with inertial and dynamic boundary conditions, proving well-posedness, existence of a global attractor, and convergence to equilibrium.

Contribution

It introduces dynamic boundary conditions into the non-isothermal viscous Cahn--Hilliard model and establishes key mathematical properties including well-posedness and long-term behavior.

Findings

Proved well-posedness for solutions with bounded energy and weak solutions.

Established the existence of a global attractor for the system.

Showed convergence of solutions to equilibrium using a Lojasiewicz--Simon inequality.

Abstract

We consider a non-isothermal modified Cahn--Hilliard equation which was previously analyzed by M. Grasselli et al. Such an equation is characterized by an inertial term and a viscous term and it is coupled with a hyperbolic heat equation. The resulting system was studied in the case of no-flux boundary conditions. Here we analyze the case in which the order parameter is subject to a dynamic boundary condition. This assumption requires a more refined strategy to extend the previous results to the present case. More precisely, we first prove the well-posedness for solutions with bounded energy as well as for weak solutions. Then we establish the existence of a global attractor. Finally, we prove the convergence of any given weak solution to a single equilibrium by using a suitable Lojasiewicz--Simon inequality.