Local existence of \(C^{\infty}\)-solution for the initial-boundary value problem of fully nonlinear wave equation (Q1059791)

Let \(\Omega \subset R^ n\) be a domain with compact \(C^{\infty}\) boundary \(\partial \Omega\), \(0<T<\infty\) and let us consider the following initial-boundary value problem: \({\mathcal L}u+F(t,x,D^ 2u)=f(t,x)\) in [0,T]\(\times \Omega\), \(u=0\) on [0,T]\(\times \partial \Omega\) and \(u(0,x)=\psi_ 0(x),\quad \partial_ tu(0,x)=\psi_ 1(x)\) in \(\Omega\), where \({\mathcal L}\) is a linear operator of the second order in t,x with strictly elliptic main part in x and with smooth coefficients \(a_{\alpha}(t,x)\). The author states a theorem on the existence of a local (in t) solution with smoothness corresponding to the data, in particular \(C^{\infty}\)-solution. Details of the proof will appear elsewhere.