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: ⎧⎨ ⎩ χt − χ +χ3 −χ −θ = 0, θt +χt + divq = 0, qt +q+∇θ = 0, (0.1) in Ω ×(0,+∞) subject to the homogeneous Neumann boundary condition for χ, ∂nχ = 0, on Γ ×(0,+∞), (0.2) and no-heat flux boundary condition for q, q · n = 0, on Γ ×(0,+∞), (0.3) and the initial conditions χ(0) = χ0, θ(0) = θ0, q(0) = q0, in Ω, (0.4) 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. © 2007 Elsevier Inc. All rights reserved.