Systems of hyperbolic conservation laws with memory

Global weak solutions of bounded variation to systems of balance laws with non-local source are constructed by the method of vanishing viscosity. Suitable dissipativeness assumptions are imposed on the source terms to assure convergence of the method. Under these hypotheses, the total variation remains uniformly bounded and integrable in time, the vanishing viscosity solutions are uniformly stable in L-1 with respect to the initial data and converge to equilibrium as t -> infinity. The motivation to study these systems is the observation that conservations laws with fading memory can be written in such form under appropriate conditions on the flux.