Attractors for a Caginalp Phase-field Model with Singular Potential

Abstract

We consider a phase field model based on a generalization of the Maxwell Cattaneo heat conduction law, with a logarithmic nonlinearity, associated with Neumann boundary conditions. The originality here, compared with previous works, is that we obtain global in time and dissipative estimates, so that, in particular, we prove, in one and two space dimensions, the existence of a unique solution which is strictly separated from the singularities of the nonlinear term, as well as the existence of the finite-dimensional global attractor and of exponential attractors. In three space dimensions, we prove the existence of a solution. AMS subject classifications: 35B40, 35B41, 35K51, 80A22, 80A20, 35Q53, 45K05, 35K55, 35G30, 92D50