Abstract: | This paper studies tho Cauchy problem of totally characteristic hyperbolic operator (1.1) in Gevrey classes, and obtains the following main result: Under the conditions (I) - (VI), if 1 ≤ s < \frac{σ}{σ-1} (σ is definded by (1.7)). then the Cauchy problem (1.1) is wellposed in B (0, T], G^s_{L²}, (R^n)); if s = \frac{σ}{σ-1}, then the Cauchy problem (1.1) is wellpooed in B (0, e], G^{\frac{σ}{σ-1}}_{L²}(R^n)) (where e > 0, small enough). |