JournalszaaVol. 25 , No. 1DOI 10.4171/zaa/1277

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

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.