Convergence to equilibrium for a parabolic–hyperbolic phase-field system with dynamical boundary condition

This paper is concerned with the well-posedness and the asymptotic behavior of solutions to the following parabolic–hyperbolic phase-field system (θ +χ)t − θ = 0, χtt + χt − χ + φ(χ)− θ = 0, (0.1) in Ω ×(0,+∞), subject to the Neumann boundary condition for θ ∂νθ = 0, on Γ ×(0,+∞), (0.2) the dynamical boundary condition for χ ∂νχ +χ +χt = 0, on Γ ×(0,+∞), (0.3) and the initial conditions θ(0) = θ0, χ(0) = χ0, χt (0) = χ1, in Ω, (0.4) where Ω is a bounded domain in R3 with smooth boundary Γ , ν is the outward normal direction to the boundary and φ is a real analytic function. In this paper we first establish the existence and uniqueness of a global strong solution to (0.1)–(0.4). Then, we prove its convergence to an equilibrium as time goes to infinity and we provide an estimate of the convergence rate. © 2006 Elsevier Inc. All rights reserved.