Hyperbolic Limit of Parabolic Semilinear Heat Equations with Fading Memory

Abstract

This paper is devoted to the comparison of two models describing heat conduction with memory, arising in the frameworks of Coleman-Gurtin and Gurtin-Pipkin. In particular, the second model entails an equation of hyperbolic type, where the dissipation is carried out by the memory term solely, and can be viewed as the limit of the first model as the coefficient $\omega$ of the laplacian of the temperature tends to zero. Results concerning the asymptotic behavior, with emphasis on the existence of a uniform attractor, are provided, uniformly in $\omega$. The attractor of the hyperbolic model is shown to be upper semicontinuous with respect to the family of attractors of the parabolic models, as $\omega$ tends to zero.