Weakly Hyperbolic Systems by Symmetrization

We prove Gevrey well posedness of the Cauchy problem for general linear systems whose principal symbol is hyperbolic and coefficients are sufficiently Gevrey regular in x and either Lipschitzian or Holderian in time. Such results date to the seminal paper of Bronshtein. Our proof is by an energy method using a pseudodifferential symmetrizer. The construction of the symmetrizer is based on a Lyapunov function for ordinary differential equations. The method yields new estimates and existence uniformly for spectral truncations and parabolic regularizations.