The Goursat problem in anisotropic Gevrey classes (Q1567128)

The authors consider the Goursat problem for a class of nonlinear equations of the type \[ D^\alpha_xu(x,y) = f \Biggl(x,y,D^Bu(x,y), \int_\Omega |D^Bu(x,y)|^2 dy\Biggr). \] This problem is a generalization of the Goursat problem for the Carrier-Kirchhoff equation \[ {\partial^2u\over\partial t^2}- \Biggl({P_0\over\rho h}+{E\over 2L\rho}\int_0^L|{\partial u\over \partial s}(s,t)|^2 \Biggr) {\partial^2u\over\partial x^2} =0, \] where \(P_0\), \(\rho\), \(h\), \(E\), \(L\) are known constants. The authors introduce a space of nonisotropic Gevrey functions (i.e. functions having different Gevrey index in different directions) and, following the ideas of \textit{C. Wagschal} [J. Math. Pures Appl., IX. Ser. 58, 309-337 (1979; Zbl 0427.35021)] and \textit{D. Gourdin} and \textit{M. Mechab} [J. Math. Pures Appl., IX. Ser. 75, No. 6, 569-593 (1996; Zbl 0869.35027)], they succeed in reducing the problem to a fixed point problem in a particular Banach algebra.