Cauchy problems for hyperbolic systems in Rn with irregular principal symbol in time and for |x|→∞

The aim of this paper is to present an approach for the study of well-posedness for diagonalizable hyperbolic systems of (pseudo)differential equations with characteristics which are not Lipschitz continuous with respect to both the time variable t (locally) and the space variables x∈Rn for |x|→∞. We introduce optimal conditions guaranteeing the well-posedness in the scale of the weighted Sobolev spaces Hs1,s2(Rn), cf. Introduction, with finite or arbitrarily small loss of regularity. We give explicit examples for ill-posedness of the Cauchy problem in the Schwartz spaces when the hypotheses on the growth for |x|→∞ fail.