Nonlinear Goursat problem in Gevrey spaces for generalized Kirchhoff equations (Q1354984)

We consider the Goursat problem for nonlinear partial differential equations, with quadratic means, of type \[ D^\alpha_xu(x,y)= f\Biggl(x,y,D^Bu(x,y),\;\int_\Omega[D^Bu(x,y)]^2dy\Biggr), \] where \(x\) is in a neighbourhood of \(0\) in \(\mathbb{R}^p\) or \(\mathbb{C}^p\), \(B\) is a finite part of the set \(\{(\gamma,\delta)\in\mathbb{Z}^p\times \mathbb{N}^q; |\gamma|+ d|\delta|\leq |\alpha|, \gamma< \alpha\}\), \(d\geq 1\). These equations generalize the Kirchhoff equation \[ D^2_tu(t,y)= a\Biggl(\int_\Omega|\nabla_y u(t,y)|^2dy\Biggr) \Delta_yu(t,y). \] With suitable assumption upon \(f\), we prove existence and uniqueness of a local solution, which is continuous with respect to \(x\) and belongs to Gevrey class with respect to \(y\), or a local solution, which is holomorphic with respect to \(x\) and belongs to Gevrey class with respect to \(y\) (with index \(d\)). The methods of proof are based on formal norms of Leray-Waelbroeck and the fixed point theorem.