2006/11/06 by Maurizio Grasselli, Grasselli, Maurizio, Hana Petzeltová +3
Computer Science · Engineering · Materials Science · #35B40 #35B41 #35R35 #80A22 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #Solidification and crystal growth phenomena #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.math/0611134
openalex publication_date 2006/11/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 χtt, χ 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.