On the time-dependent Cattaneo law in space dimension one

We consider the one-dimensional wave equation epsilon u(tt) - u(xx) + [1 + ef' (u)]u(t) + f(u) = h where epsilon = epsilon(t) is a decreasing function vanishing at infinity, providing a model for heat conduction of Cattaneo type with thermal resistance decreasing in time. Within the theory of processes on time-dependent spaces, we prove the existence of an invariant time-dependent attractor, which converges in a suitable sense to the attractor of the classical Fourier equation u(t) - u(xx) + f(u) = h formally arising in the limit t -> infinity. (C) 2015 Elsevier Inc. All rights reserved.