# American Institute of Mathematical Sciences

December  2006, 5(4): 827-838. doi: 10.3934/cpaa.2006.5.827

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

 1 Dipartimento di Matematica "F. Brioschi", Politecnico di Milano, I-20133 Milano 2 Mathematical Institute AV ČR, Žitná 25, 115 67 Praha 1 3 Dipartimento di Matematica "F.Casorati", Università di Pavia, Via Ferrata, 1, I-27100 Pavia

Received  December 2005 Revised  June 2006 Published  September 2006

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.
Citation: M. Grasselli, Hana Petzeltová, Giulio Schimperna. Convergence to stationary solutions for a parabolic-hyperbolic phase-field system. Communications on Pure & Applied Analysis, 2006, 5 (4) : 827-838. doi: 10.3934/cpaa.2006.5.827
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