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Existence of global solutions for nonlinear wave equations. (English) Zbl 0548.35078

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 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 \(\lambda_ i,\lambda_{i,j}\).
Reviewer: J.Wang

MSC:

35L70 Second-order nonlinear hyperbolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)

Citations:

Zbl 0405.35056
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References:

[1] Chao-Hao, G.: Comm. Pure Appl. Math., 33, 727-737 (1980). · Zbl 0475.58005 · doi:10.1002/cpa.3160330604
[2] John, F.: ibid., 34, 29-51 (1981). · Zbl 0453.35060 · doi:10.1002/cpa.3160340103
[3] Klainerman, S.: ibid., 33, 43-101 (1980). · Zbl 0405.35056 · doi:10.1002/cpa.3160330104
[4] Glassey, R. T.: Math. Z., 178, 233-261 (1981). · Zbl 0451.35039 · doi:10.1007/BF01262042
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