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We consider a class of weakly hyperbolic systems of first-order, nonlinearPDEs. Weak hyperbolicity means here that the principal symbol of the system hasa crossing of eigenvalues, and is not uniformly diagonalizable. We prove thewell-posedness of the Cauchy problem in the Gevrey regularity for all Gevreyindices greater than $1/2$. The proof is based on the construction of asuitable approximate symmetrizer of the principal symbol and an energy estimatein Gevrey spaces. We discuss both the generality of the assumption on thestructure of the principal symbol and the sharpness of the lower bound of theGevrey index.

On hyperbolicity and Gevrey well-posedness. Part three: a class of weakly hyperbolic systems