A nonlocal phase-field system with inertial term

Abstract

Abstract. We study a phase-field system where the energy balance equation has the standard (parabolic) form, while the kinetic equation ruling the evolution of the order parameter χ is a nonlocal and nonlinear second-order ODE. The main features of the latter equation are a space convolution term which models long-range interactions of particles and a singular configuration potential that forces χ to take values in (−1, 1). We first prove the global existence and uniqueness of a regular solution to a suitable initial and boundary value problem associated with the system. Then, we investigate its long time behavior from the point of view of ω-limits. In particular, using a nonsmooth version of the Lojasiewicz-Simon inequality, we show that the ω-limit of any trajectory contains one and only one stationary solution, provided that the configuration potential in the kinetic equation is convex and analytic.

Finite-Dimensional Global Attractor for a Nonlocal Phase-Field System

“…also [6] for the nonlocal Allen-Cahn equation). A similar issue is analyzed in the more recent contribution [35], where equation (1.9) has an additional relaxation term of the form ε χ tt and W is a singular potential (though not of type (1.3)). However, none of the above results is concerned with the existence of a global attractor even though this is a rather natural feature of phase-field dynamics (cf.…”

Non-standard applications of the Łojasiewicz-Simon theory: Stabilization to equilibria of solutions to phase-field models

“…All the convergence results in Theorems 3.2-3.4 can be adapted to the case when the ∂ t χ in the equations for the order parameter is replaced by the "hyperbolic" operator ε∂ tt χ + ∂ t χ (see [21], [22], [23]). Here the basic tool is a variant of the Łojasiewicz-Simon theory for hyperpolic problems with damping developed by Haraux and Jendoubi [25].…”

Regularity and separation from potential barriers for a non-local phase-field system

“…Phase-field systems with singular potentials and the separation property were analyzed, in the non-degenerate situation, in [14,15]. In the latter paper, a non-local (in space) system with inertial term was studied with the assumption that the initial function is already separated from the pure phases.…”