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 systemequation(0.1){(θ+χ)t−Δθ=0,χtt+χt−Δχ+ϕ(χ)−θ=0, in Ω×(0,+∞)Ω×(0,+∞), subject to the Neumann boundary condition for θequation(0.2)∂νθ=0,on Γ×(0,+∞), the dynamical boundary condition for χequation(0.3)∂νχ+χ+χt=0,onΓ×(0,+∞), and the initial conditionsequation(0.4)θ(0)=θ0,χ(0)=χ0,χt(0)=χ1,in Ω, where Ω   is a bounded domain in R3R3 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.