Uniform attractors for a phase-field model with memory and quadratic nonlinearity

A phase-field system with memory which describes the evolution of both the temperature variation $\theta$ and the phase variable $\chi$ is considered. This thermodynamically consistent model is based on a linear heat conduction law of Coleman-Gurtin type. Moreover, the internal energy linearly depends both on the present value of $\theta$ and on its past history, while the dependence on $\chi$ is represented through a function with quadratic nonlinearity. A Cauchy-Neumann initial and boundary value problem associated with the evolution system is then formulated in a history space setting. This problem is shown to generate a non-autonomous dynamical system which possesses a uniform attractor. In the autonomous case, the attractor has finite Hausdorff and fractal dimensions whenever the internal energy linearly depends on $\chi$.