Zeitschrift für Analysis und ihre Anwendungen


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Volume 25, Issue 1, 2006, pp. 51–72
DOI: 10.4171/ZAA/1277

Published online: 2006-03-31

Long Time Behavior of Solutions to the Caginalp System with Singular Potential

Maurizio Grasselli[1], Hana Petzeltová[2] and Giulio Schimperna[3]

(1) Politecnico di Milano, Italy
(2) Czech Academy of Sciences, Prague, Czech Republic
(3) Università di Pavia, Italy

We consider a nonlinear parabolic system which governs the evolution of the (relative) temperature $\teta$ and of an order parameter $\chi$. This system describes phase transition phenomena like, e.g., melting-solidification processes. The equation ruling $\chi$ is characterized by a singular potential $W$ which forces $\chi$ to take values in the interval $[-1,1]$. We provide reasonable conditions on $W$ which ensure that, from a certain time on, $\chi$ stays uniformly away from the pure phases $1$ and $-1$. Combining this separation property with the {\L}ojasiewicz-Simon inequality, we show that any smooth and bounded trajectory uniformly converges to a stationary state and we give an estimate of the decay rate.

Keywords: Phase-field models, maximal monotone operators, comparison principle, asymptotic behavior, Lojasiewicz-Simon inequality

Grasselli Maurizio, Petzeltová Hana, Schimperna Giulio: Long Time Behavior of Solutions to the Caginalp System with Singular Potential. Z. Anal. Anwend. 25 (2006), 51-72. doi: 10.4171/ZAA/1277