On the radius of spatial analyticity for semilinear symmetric hyperbolic systems

We study the problem of propagation of analytic regularity for semi-linear symmetric hyperbolic systems. We adopt a global perspective and we prove that if the initial datum extends to a holomorphic function in a strip of radius (=width) $\epsilon_0$, the same happens for the solution $u(t,\cdot)$ for a certain radius $\epsilon(t)$, as long as the solution exists. Our focus is on precise lower bounds on the spatial radius of analyticity $\epsilon(t)$ as $t$ grows. \par We also get similar results for the Schr\"odinger equation with a real-analytic electromagnetic potential.