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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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