Hölder continuity in time for SG hyperbolic systems (Q2425379)

The authors detail with the Cauchy problem for the first-order hyperbolic equations \[ \partial_t u(t,x)= K(t,x,D)u(t,x),\;t\in[0,T],\;x\in\mathbb{R}^n, \] \[ u|_{t=0}= u_0(x), \] where \(K\) is a matrix of pseudo-differential operators of order one with symbol \(k(t,x,\xi)\) defind in \([0, T]\times\mathbb{R}^n\times \mathbb{R}^n\), analytic in \(\xi\), in Gevrey class of index \(\sigma\) in \(x\) and \(\mu\)-Hölder continuous with respect to the time variable \(t\). They showed that if \(k(t,x,\xi)\) is symmetrizable and \(1<\sigma<{1\over 1-\mu}\), \(0<\mu< 1\) holds, the above Cauchy problem is well-posed in Gevrey class of index \(\sigma\).