Communications on Pure and Applied Analysis (CPAA)

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

Pages: 827 - 838, Volume 5, Issue 4, December 2006      doi:10.3934/cpaa.2006.5.827

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M. Grasselli - Dipartimento di Matematica "F. Brioschi", Politecnico di Milano, I-20133 Milano, Italy (email)
Hana Petzeltová - Mathematical Institute AV ČR, Žitná 25, 115 67 Praha 1, Czech Republic (email)
Giulio Schimperna - Dipartimento di Matematica "F.Casorati", Università di Pavia, Via Ferrata, 1, I-27100 Pavia, Italy (email)

Abstract: 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.

Keywords:  Phase-field models, convergence to stationary solutions, Lojasiewicz-Simon inequality.
Mathematics Subject Classification:  35B40, 35Q99, 80A22.

Received: December 2005;      Revised: June 2006;      Available Online: September 2006.