Long time behavior of solutions to the Caginalp system with singular potential ∗ Maurizio Grasselli Dipartimento di Matematica, Politecnico di Milano Via Bonardi, 9 I-20133 Milano, Italy maugra@mate.polimi.it HanaPetzeltov´a † Mathematical Institute AS CR ˇ Zitn´a,25 CZ-115 67, Praha 1, Czech Republic petzelt@math.cas.cz Giulio Schimperna ‡ Dipartimento di Matematica, Universit`a di Pavia Via Ferrata, 1 I-27100 Pavia, Italy giulio@dimat.unipv.it Abstract We consider a nonlinear parabolic system which governs the evolution of the (rel- ative) temperature ϑ and of an order parameter χ . This system describes phase transition phenomena like, e.g., melting-solidification processes. The equation ruling χ is characterized by a singular potential W which forces χ to take values in the interval [−1, 1]. We provide reasonable conditions on W which ensure that, from a certain time on, χ stays uniformly away from the pure phases 1 and −1. Combining this separation property with the  Lojasiewicz-Simon inequal- ity, we show that any smooth and bounded trajectory uniformly converges to a stationary state and we give an estimate of the decay rate. * This work was partially supported by the Italian MIUR PRIN Research Projects Modellizzazione Matematica ed Analisi dei Problemi a Frontiera Libera and Aspetti Teorici e Applicativi di Equazioni a Derivate Parziali, and by the Italian MIUR FIRB Research Project Analisi di Equazioni a Derivate Parziali, Lineari e Non Lineari: Aspetti Metodologici, Modellistica, Applicazioni † The work of H.P. was supported by the Grant A1019302 of GA AV ˇ CR ‡ The work of G.S. was partially supported by the HYKE Research Training Network 1