Abstract. In this paper we investigate the initial and initial-boundary value problems for strictly hyperbolic balance laws with time-evolution of flux and source. Such nonlinear balance laws arise in, for instance, gas dynamics equations in time-dependent ducts and nozzles, shallow water equations, lanes-changing model in traffic flow and Einstein's field equations in a spherically symmetric spacetime. To account for the time dependence of flux and source, we introduce the perturbed Riemann and boundary Riemann problems. Such Riemann problems have unique solutions within elementary waves and an additional family of waves. Based on the work of [12,13], a new version of Glimm scheme is introduced and its stability is established by modified interaction estimates. Finally, the existence of global entropy solutions is achieved by showing the consistency of scheme, the weak convergence of source term and the entropy inequalities.

MSC: 35L60 35L65 35L67Keywords: Quasi-linear wave equations Hyperbolic integro-differential systems Nonlinear balance laws Entropy solutions Cauchy problem Riemann problem Perturbed Riemann problem Generalized Glimm scheme This research explores the Cauchy problem for a class of quasilinear wave equations with time dependent sources. It can be transformed into the Cauchy problem of hyperbolic integro-differential systems of nonlinear balance laws. We introduce the generalized Glimm scheme in new version and study its stability which is proved by Glimm-type interaction estimates in a dissipativity assumption. The generalized solutions to the perturbed Riemann problems, the building blocks of generalized Glimm scheme, are constructed by Riemann problem method modeled on the source free equations. The global existence for the Lipschitz continuous solutions and weak solutions to the systems is established by the consistency of scheme and the weak convergence of source. Finally, the weak solutions are also the entropy solutions which satisfy the entropy inequality.