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Convergence to stationary solutions for a parabolic-hyperbolic phase-field system

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  • A parabolic-hyperbolic nonconserved phase-field model is here analyzed. This is an evolution system consisting of a parabolic equation for the relative temperature $\theta$ which is nonlinearly coupled with a semilinear damped wave equation governing the order parameter $\chi$. The latter equation is characterized by a nonlinearity $\phi(\chi)$ with cubic growth. Assuming homogeneous Dirichlet and Neumann boundary conditions for $\theta$ and $\chi$, we prove that any weak solution has an $\omega$-limit set consisting of one point only. This is achieved by means of adapting a method based on the Łojasiewicz-Simon inequality. We also obtain an estimate of the decay rate to equilibrium.
    Mathematics Subject Classification: 35B40, 35Q99, 80A22.

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