Local existence for solutions of fully non‐linear wave equations

Let Ω be a domain in ℝ n and let m ϵ ℕ; be given. We study the initial‐boundary value problem for the equation\documentclass{article}\pagestyle{empty}\begin{document}$$ F{\rm (}t,x,\bar D{\textstyle{{2m} \over x}}u{\rm (}t,x{\rm ),}\bar D{\textstyle{m \over x}}\mathop \partial \nolimits_t u{\rm (}t,x{\rm ),}\mathop \partial \nolimits_t^2 u{\rm (}t,x{\rm )) = }f{\rm (}t,x{\rm )} $$\end{document} with a homogeneous Dirichlet boundary condition; here u is a scalar function, \documentclass{article}\pagestyle{empty}\begin{document}$ \bar D_x^m u: = (\partial _x^\alpha u)_{|\alpha | \le m} $\end{document} and certain restrictions are made on F guaranteeing that energy estimates are possible. We prove the existence of a value of T >0 such that a unique classical solution u exists on [0, T ]×Ω. Furthermore, we show that T → ∞ if the data tend to zero.