An Exponential Attractor for a Two-Temperature Phase Transition Model
Brice Landry DOUMBE BANGOLA
Departement de Mathematiques et Informatique, Faculte des Sciences (FS), Universite des Sciences et Techniques de Masuku, Franceville, Gabon.
Mohamed Ali IPOPA *
Departement Sciences Generales de l'Ingenieur, Ecole Polytechnique de Masuku (EPM), Universite des Sciences et Techniques de Masuku, Franceville, Gabon.
Armel ANDAMI OVONO
Departement de Mathematiques, Ecole Normale Superieure (ENS), Libreville, Gabon.
*Author to whom correspondence should be addressed.
Abstract
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.
Keywords: Transition model, two temperatures, global attractor, exponential attractor, Hausdor 's dimension, fractal dimension