Title: Exponential attractors for a phase-field model with memory and quadratic nonlinearity

Authors: Stefania Gatti, Maurizio Grasselli and Vittorino Pata

Issue: Volume 53 (2004), Issue 3, 719-754

Abstract: We consider a phase-field system with memory effects. This model consists of an integrodifferential equation of parabolic type describing the evolution of the (relative) temperature $\theta$, and depending on its past history. This equation is nonlinearly coupled through a function $\lambda$ with a semilinear parabolic equation governing the order parameter $\chi$. The state variables $\theta$ and $\chi$ are subject to Neumann homogeneous boundary conditions. The model becomes an infinite-dimensional dynamical system in a suitable phase-space by introducing an additional variable $\eta$ accounting for the (integrated) past history of the temperature. The evolution of $\eta$ is thus ruled by a first-order hyperbolic equation. Giorgi, Grasselli, and Pata proved that the obtained dynamical system possesses a universal attractor $\mathcal{A}$, which has finite fractal dimension provided that the coupling function $\lambda$ is linear. Here we prove, as main result, the existence of an exponential attractor $\mathcal{E}$ which entails, in particular, that $\mathcal{A}$ has finite fractal dimension when $\lambda$ is nonlinear with quadratic growth. Since the so-called squeezing property does not work in our framework, we cannot use the standard technique to construct $\mathcal{E}$. Instead, we take advantage of a recent result due to Efendiev, Miranville, and Zelik. The present paper contains, to the best of our knowledge, the first example of exponential attractor for an infinite-dimensional dynamical system with memory effects. Also, the approach introduced here can be adapted to other dynamical systems with similar features.