Cauchy problems for hyperbolic systems in $R^n$ with irregular principal symbol in time and for $|x| \rightarrow \infty$.

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 $x\in \R^n$ for $|x|\to \infty$. We introduce optimal conditions guaranteeing the well-posedness in the scale of the weighted Sobolev spaces $H^{s_1,s_2}(\R^n),$ cf. Introduction, with finite or arbitrarily small loss of regularity. We give explicit examples for ill-possedness of the Cauchy problem in the Schwartz spaces when the hypotheses on the growth for $|x|\to \infty$ fail.