Uniqueness and Continuous Dependence for Systems of Balance Laws with Dissipation

We consider the Cauchy problem for the × strictly hyperbolic system of balance laws in one space dimensioneach characteristic field being genuinely nonlinear or linearly degenerate. We prove the existence of a unique Lipschitz flow of entropic solutions defined for with sufficiently small total variation; under the assumption that the source term is dissipative, the flow turns out to be a semigroup defined for all ≥ 0. Moreover, each trajectory of the flow is the unique solution of a Cauchy problem for a generalized differential equation