Log-Lipschitz regularity for SG-hyperbolic systems

We consider the Cauchy problem for linear and quasilinear symmetrizable hyperbolic systems with coefficients depending on time and space, not smooth in $t$ and growing at infinity with respect to $x.$ We discuss well posedness in weighted Sobolev spaces, showing that the non-Lipschitz regularity in $t$ has an influence not only on the loss of derivatives of the solution but also on its behaviour for $|x| \rightarrow \infty.$ We provide examples to prove that the latter phenomenon cannot be avoided.