An Exponential Attractor for a Two-Temperature Phase Transition Model

In this work, we investigate the finite dimensionality of an attractor of a two-temperature Caginalptype system for heat conduction. In order to prove that the global attractor is of finite dimension, we can use the volume contraction method, show the existence of an inertial manifold or an exponential attractor. The volume contraction method is not applicable because it requires a certain differentiability of the associated semigroup, which is not possible to obtain for our system. Similarly, the construction of an inertial manifold relies on the so-called spectral gap condition, which is a very restrictive condition. For all these reasons, we show that the global attractor of the system is of finite fractal dimension by proving that the system has an exponential attractor.