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

Abstract

We consider a nonlinear parabolic system which governs the evolution of the (relative) temperature $\vartheta$ 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 Łojasiewicz–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.