Cauchy problems for hyperbolic systems in 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 for x→∞. We introduce optimal conditions guaranteeing the well-posedness in the scale of the weighted Sobolev spaces , 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.