Nonlinear hyperbolic Cauchy problems in Gevrey classes (Q5955014)

The authors consider the quasilinear Cauchy problem \[ \sum_{ |\alpha|\leq m}a_\alpha (t,x,D^\beta_{t,x} u)D^\alpha_{t,x} u=f(t,x, D^\beta_{t,x}u), \] \[ D^j_t u|_{t=0}=0,\;0\leq j<m, \] where in the nonlinearities \(|\beta |\leq m'\leq m-1\). The main assumptions are the following: the coefficients \(a_\alpha\), \(|\alpha |=m\), are Hölder functions of order \(p\), \(0<p<1\), with respect to the time variable \(t\) and the operator is hyperbolic with constant multiplicity \(\leq r\). Assuming further \(m'\leq m-\min \{r,2\}\), the authors are then able to conclude well-posedness in the Gevrey classes \(G^s\) for \(1<s< r/(r-p)\), that is a solution exists in \(G^s\) if \(a_\alpha\) and \(f\) have the same regularity with respect to the \(x\)-variables. This extends several known results, for example in the strictly hyperbolic case one obtains \(s<1/(1-p)\) as in \textit{F. Colombini}, \textit{E. De Giorgi} and \textit{S. Spagnolo} [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4) 6, 511-559 (1979; Zbl 0417.35049)]. The limit case \(p=1\) gives the standard well-posedness in Gevrey classes for \(s<r/(r-1)\).