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Convergence to equilibrium for a parabolic-hyperbolic phase-field system with dynamical boundary condition. (English) Zbl 1154.35329
Summary: This paper is concerned with the well-posedness and the asymptotic behavior of solutions to the following parabolic-hyperbolic phase-field system
\[ \begin{cases}(\theta+\chi)_t -\Delta \theta = 0\\ \chi_{tt}+\chi_t -\Delta_{\chi}+ \varphi(\chi)-\theta = 0\end{cases}\tag{1} \] in \(\Omega \times (0,+\infty)\), subject to the Neumann boundary condition for \(\theta\) \[ \partial_{\nu}\theta = 0, \quad \text{on } \Gamma \times (0, +\infty)\tag{2} \] the dynamical boundary condition for \(\chi\)
\[ \partial_{\nu}\chi + \chi +\chi_t=0,\quad \text{on } \Gamma \times (0,+\infty)\tag{3} \] and the initial
\[ \theta(0)=\theta_0, \quad \chi(0)=\chi_0,\quad \chi_t(0)=\chi_1,\quad \text{in } \Omega,\tag{4} \] where \(\Omega\) is a bounded domain in \(\mathbb R^3\) with smooth boundary \(\Gamma , \nu \) is the outward normal direction to the boundary and \(\varphi\) is a real analytic function. In this paper we first establish the existence and uniqueness of a global strong solution to \((1)--(4)\). Then, we prove its convergence to an equilibrium as time goes to infinity and we provide an estimate of the convergence rate.

MSC:
35G30 Boundary value problems for nonlinear higher-order PDEs
35K55 Nonlinear parabolic equations
35L70 Second-order nonlinear hyperbolic equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B40 Asymptotic behavior of solutions to PDEs
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