Existence of global solutions for nonlinear wave equations (Q799869)

The author considers nonlinear wave equations of the following type: \((1.1)\quad\square u+F(u,Du,D_ xDu)=0,\) for \(t\in [0,\infty)\), \(x\in R^ n\), with the initial data \((1.2)\quad u(0,x)=\phi (x),\quad (1.3)\quad (\partial u/\partial t)(0,x)=\chi (x)\quad for\quad x\in R^ n.\) Here the symbols \(D_ x\) and D denote \((\partial /\partial x_ 1,...,\partial /\partial x_ n)\) and \((\partial /\partial t,D_ x)\) respectively, and \(\square\) denotes the wave operator. Suppose that the function F of (1.1) is a function of variables \(\xi =(\lambda;\lambda_ i\), \(i=0,...,n\); \(i,j=0,...,n\), \(i+j>0)\) and it is of class \(C^{\infty}\) in a neighborhood of the origin \(\xi =0\) and \((A)\quad F(0)=(\partial F/\partial\xi )(0)=0.\) The main result: Suppose that the space dimension n is greater than or equal to 12 and that condition (A) is satisfied. Then there exist an integer N and a small constant \(\eta >0\) such that for any initial data satisfying \(\|\phi \|_{L_ 1,N}+\|\psi \|_{L_ 1,N}<\eta\) and \(\|\phi \| =_{L_ 2,N}+\|\psi \|_{L_ 2,N}<\eta,\) the problem (1) has a unique solution in \(C^{\infty}([0,\infty)\times R^ n).\) As remarked by the author, the problem considered in this paper differs from that of \textit{S. Klainerman} [Commun. Pure Appl. Math. 33, 43-101 (1980; Zbl 0405.35056)] in the point that F in this paper depends on \(\lambda\) as well as \(\lambda_ i,\lambda_{i,j}\).