On the time-dependent Cattaneo law in space dimension one

We consider the one-dimensional wave equationεutt-uxx+[1+εf′(u)]ut+f(u)=hεutt-uxx+[1+εf′(u)]ut+f(u)=hwhere ε=ε(t)ε=ε(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 equationut-uxx+f(u)=hut-uxx+f(u)=hformally arising in the limit t→∞t→∞.