An extended Cauchy-Kovalevskaya problem and its solution in associated spaces
The authors are interested in solving operator equations of the form \[ u(t,x)= F_1(t,x,u)+ \int^t_0 F_2(\tau, x,u,\nabla_xu)\,d\tau. \] Both operators are supposed to be associated to an (elliptic) operator \(G\), i.e., \(Gu= 0\) implies \(G(F_ju)= 0\) for each fixed \(t\). The ellipticity of \(G\) with time-independent coefficients is reasonable because the authors assume interior estimates for the solutions of \(Gu= 0\). Solutions of the above operator equation possess the property of conical evolution. This allows to define a suitable Banach space. Some Lipschitz properties of \(F_1\) and \(F_2\) guarantee the existence of a unique solution in this Banach space. The solutions exist locally in \(t\).
scientific article; zbMATH DE number 1925968