Nonlinear weakly hyperbolic equations in Gevrey classes (Q2704042)

The author considers a quasilinear weakly hyperbolic equation, with characteristics of constant multiplicity \(r\). The coefficients are assumed to be of Hölder class \(C^s\), \(0<s<1\), with respect to the time variable. Under these conditions, well-posedness of the Cauchy problem is proved for data in the Gevrey class \(G^\sigma\), with \(1<\sigma\leq r/(r-s)\). In the strictly hyperbolic case \(r=1\) one recaptures the result of \textit{F. Colombini}, \textit{E. De Giorgi} and \textit{S. Spagnolo} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 6, 511-559 (1979; Zbl 0417.35049)] stating well-posedness for \(\sigma<1/(1-s)\). We also recall that, in the case of smooth coefficients, well-posedness is granted for \(\sigma<r/(r-1)\), corresponding to the limit case \(s=1\). The proof uses the nonlinear microlocal techniques of \textit{M. Cicognani} and \textit{L. Zanghirati} [Bull. Sci. Math., 123, 413-435 (1999; Zbl 0940.35142)].NEWLINENEWLINEFor the entire collection see [Zbl 0958.00030].