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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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##### References:
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