The Log-effect for 2 by 2 hyperbolic systems

In the present paper we are interested to extend the Log-effect from wave equations with time-dependent coefficients to 2 by 2 strictly hyperbolic systems ∂tU−A(t)∂xU=0. Besides the effects of oscillating entries of the matrix A=A(t) and interactions between the entries of A we have to take into consideration the system character itself. We will prove by tools from phase space analysis results about H∞ well- or ill-posedness. The precise loss of regularity is of interest. Finally, we discuss the cone of dependence property.