A First Order Inhomogeneous Quasilinear Hyperbolic Cauchy Problem Of The Existence Of Classical Solutions As A Whole

The first order quasilinear hyperbolic systems with two independent variables of-ten arise in many science fields such as Physics. Generally speaking, the global classical solutions to Cauchy problems for the first order quasilinear hyperbolic systems only ex-ist in a short time even for smooth enough or small enough initial data. So people naturally raise the problem:when does a global classical solution exist? Hence, there are theoretical and practical needs to study the existence of global classical solutions.In this paper, we study the case that the inhomogeneous term is the general form of F(t,x). First of all, when the system is linearly degenerate and the inhomoge-neous term is independent of t, we prove the existence of global classical solutions to Cauchy problem for general first order inhomogeneous quasilinear hyperbolic sys-tems for small enough L1norm of inhomogeneous term and first order derivative of initial value. We also demonstrate the global existence of classical solutions to Cauchy problem for general first order inhomogeneous quasilinear hyperbolic systems in the hypothesis of weakly linearly degenerate. The formulas on the decomposition of waves play an important role in our discussion. We first give the refine formulas on the de-composition of waves under the assumption of linearly degenerate and weakly linearly degenerate respectively, when the inhomogeneous term is the general form of F(t. x). By these formulas, we obtain the existence of global classical solutions to Cauchy prob-lem for general first order inhomogeneous quasilinear hyperbolic systems briefly.