Zeitschrift für Analysis und ihre Anwendungen

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

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

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

(1) Dipartimento di Matematica, Politecnico di Milano, Via Bonardi, 9, 20133, Milano, Italy
(2) Mathematical Institute, Czech Academy of Sciences, Zitna 25, 115 67, Prague 1, Czechia
(3) Dipartimento di Matematica, Università di Pavia, Via Ferrata 1, 27100, 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