JournalszaaVol. 25, No. 1pp. 51–72

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

  • Hana Petzeltová

    Czech Academy of Sciences, Prague, Czech Republic
  • Maurizio Grasselli

    Politecnico di Milano, Italy
  • Giulio Schimperna

    Università di Pavia, Italy
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Abstract

We consider a nonlinear parabolic system which governs the evolution of the (relative) temperature \teta\teta and of an order parameter χ\chi. This system describes phase transition phenomena like, e.g., melting-solidification processes. The equation ruling χ\chi is characterized by a singular potential WW which forces χ\chi to take values in the interval [1,1][-1,1]. We provide reasonable conditions on WW which ensure that, from a certain time on, χ\chi stays uniformly away from the pure phases 11 and 1-1. 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.

Cite this article

Hana Petzeltová, Maurizio Grasselli, Giulio Schimperna, Long Time Behavior of Solutions to the Caginalp System with Singular Potential. Z. Anal. Anwend. 25 (2006), no. 1, pp. 51–72

DOI 10.4171/ZAA/1277