Longterm dynamics of a conserved phase‐field system with memory

Abstract

We consider a phase‐field model based on hereditary constitutive equations for the internal energy and the heat flux and on the assumption that the spatial average of the order parameter χ is conserved. This model consists of a parabolic integrodifferential equation for the (relative) temperature ϑ coupled with a nonlinear fourth‐order evolution equation for χ. We first show that the obtained system is indeed a nonautonomous dynamical system, provided that the phase space accounts for the past history of ϑ and appropriate boundary conditions are given. Then we establish the existence of an absorbing set, uniformly with respect to a certain class of heat source terms. Finally, we prove that, under suitable assumptions, our dissipative dynamical system possesses a uniform attractor of finite Hausdorff and fractal dimensions.