Convergence to equilibrium for a parabolic-hyperbolic phase field model with Cattaneo heat flux law

In this paper we consider the well-posedness and the asymptotic behavior of solutions to the following parabolic-hyperbolic phase field system:(0.1){χt−Δχ+χ3−χ−θ=0,θt+χt+divq=0,qt+q+∇θ=0, in Ω×(0,+∞) subject to the homogeneous Neumann boundary condition for χ,(0.2)∂nχ=0,onΓ×(0,+∞), and no-heat flux boundary condition for q,(0.3)q⋅n=0,onΓ×(0,+∞), and the initial conditions(0.4)χ(0)=χ0,θ(0)=θ0,q(0)=q0,inΩ, where Ω⊂R3 is a bounded domain with a smooth boundary Γ and n is the outward normal direction to the boundary. 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.