We consider a nonlinear parabolic system which governs the evolution of the (relative) temperature T and of an order parameter p. This system describes phase transition phenomena like, e.g., melting-solidification processes. The equation ruling p is characterized by a singular potential W which forces p to take values in the interval [-1,1]. We provide reasonable conditions on W which ensure that, from a certain time on, p stays uniformly away from the pure phases 1 and -1. Combining this separation property with the Lojasiewicz-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.
Long time behavior of solutions to the Caginalp system with singular potential
SCHIMPERNA, GIULIO FERNANDO
2006-01-01
Abstract
We consider a nonlinear parabolic system which governs the evolution of the (relative) temperature T and of an order parameter p. This system describes phase transition phenomena like, e.g., melting-solidification processes. The equation ruling p is characterized by a singular potential W which forces p to take values in the interval [-1,1]. We provide reasonable conditions on W which ensure that, from a certain time on, p stays uniformly away from the pure phases 1 and -1. Combining this separation property with the Lojasiewicz-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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
