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Global solutions of nonlinear hyperbolic equations for small initial data. (English) Zbl 0612.35090
The following system of N partial differential equations with \(n+1\) independent variables (n odd, \(n\geq 3)\) is considered: (1) \(\square u=f_ k(u,\partial u,\partial^ 2u)\), where u indicates N variables \(u^ A\) \((A=1,...,N)\), \(\partial^ ku=\partial^ ku^ A/\partial x^{\mu_ 1}...\partial^{\mu_ k}\) \((\mu_ 1,...,\mu_ k=0,...n)\), \(\square =\eta^{\mu \nu}\partial^ 2/\partial x^{\mu}\partial x^{\nu}\), where \(\eta\) is the Minkowski metric in \(R^{n+1}\), \(\eta =diag(-1,+1,...,+1)\); f(u,v,w) indicates the N functions \(f^ A(u,v,w)\), \((A=1,...,N)\) of class \(C^{\infty}\), defined in the domain \({\mathcal U}\times R^{N(n+1)(n+2)/2}\); \({\mathcal U}\) is an open set of \(R^ N\times R^{N(n+1)}\) containing the point (0,0,0), there exist \(f(0,0,0)=0\), \(f'(0,0,0)=0\), \(f^ A(u,v,w)=\alpha_ B^{A\mu \nu}(u,v)w^ B_{\mu \nu}+\beta^ A(u,v)\); \(\alpha_ B^{A_{\mu \nu}}=\delta^ A_ B\alpha^{\mu \nu}\) so that system (1) is almost diagonal; system (1) is supposed to be hyperbolic.
For the system (1) the global Cauchy problem with the initial values (2) \(u|_{x^ 0=0}=Q\), \(\partial u/\partial x_ 0|_{x_ 0=0}=P\) (the initial values are sufficiently small) is studied. Under some hypotheses and in a convenient functional domain the existence of a unique global solution to the Cauchy problem (1), (2) is proved and its asymptotic behaviour is studied.

MSC:
35L70 Second-order nonlinear hyperbolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B40 Asymptotic behavior of solutions to PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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