On the nonlinear Cauchy problem

The paper deals with the Cauchy problem for quasilinear higher-order hyperbolic equations. For different cases, using a unified procedure, local existence and uniqueness of solutions as well as propagation-of-regularity results are obtained. In particular, in the case when the characteristic equation has only real roots, each having constant multiplicity (call this case CM), the well-posedness of the Cauchy problem in $C^{\infty}$ and in Sobolev-Gevrey spaces is proved. The authors also consider the case when the equation is strictly hyperbolic and the coefficients of the principal part $a_{\alpha}$ are Log-Lipschitz in the variable $t$ or only $C^{1}$ in $(0,T)$, obey the estimate $|\partial_{t}a_{\alpha}|\leq c t^{-1}$, $|\alpha|=m$, and are smooth in the other variables, as well as the case (CM) under the assumption that the coefficients $a_{\alpha},$ $|\alpha|=m$, are only Hölder continuous with exponent less than 1 in the variable $t$ and smooth in the other variables.