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Convergence to stationary solutions for a parabolic-hyperbolic phase-field system. (English) Zbl 1134.35017
The authors deal with the evolution system for the pair \((\theta,X)\)
\[ \begin{aligned} (\theta+ \lambda(X))_t- \Delta\theta=f &\quad\text{in }\Omega,\\ \varepsilon X_{tt}+ X_t-\Delta X+ X+\varphi(X)- \lambda'(X)\theta=0 &\quad\text{in }\Omega,\end{aligned}\tag{1} \] where \(\Omega\subset\mathbb{R}^3\) is a bounded domain with smooth boundary \(\partial\Omega\). The main goal of the authors is the convergence of the dynamics generated by (1). More precisely, they show that, if \(\varphi\) is real analytic and satisfies suitable growth and coercivity assumptions, then any (weak) solution converges to a single stationary state.

MSC:
35B40 Asymptotic behavior of solutions to PDEs
80A22 Stefan problems, phase changes, etc.
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