Mathematics – Analysis of PDEs
Scientific paper
2006-11-06
Mathematics
Analysis of PDEs
Scientific paper
We consider a differential model describing nonisothermal fast phase separation processes taking place in a three-dimensional bounded domain. This model consists of a viscous Cahn-Hilliard equation characterized by the presence of an inertial term $\chi_{tt}$, $\chi$ being the order parameter, which is linearly coupled with an evolution equation for the (relative) temperature $\teta$. The latter can be of hyperbolic type if the Cattaneo-Maxwell heat conduction law is assumed. The state variables and the chemical potential are subject to the homogeneous Neumann boundary conditions. We first provide conditions which ensure the well-posedness of the initial and boundary value problem. Then, we prove that the corresponding dynamical system is dissipative and possesses a global attractor. Moreover, assuming that the nonlinear potential is real analytic, we establish that each trajectory converges to a single steady state by using a suitable version of the Lojasiewicz-Simon inequality. We also obtain an estimate of the decay rate to equilibrium.
Grasselli Maurizio
Petzeltová Hana
Schimperna Giulio
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