The Cauchy problem for quasilinear SG-hyperbolic systems

We study the Cauchy problem for a class of quasilinear hyperbolic systems with coefficients depending on (t, x) is an element of [0, T] x R-n and presenting a linear growth for vertical bar x vertical bar -> infinity. We prove well-posedness in the Schwartz space S(R-n). The result is obtained by deriving an energy estimate for the solution of the linearized problem in some weighted Sobolev spaces and applying a fixed point argument. (c) 2007 WTLEY-VCH Verlag GmbH & Co. KGaA, Weinheim.