Abstract :
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.
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