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 \(u^ A\) \((A=1,\dots,N)\), \(\partial^ ku=\partial^ ku^ A/\partial x^{\mu_ 1}\dots\partial^{\mu_ k}\) \((\mu_ 1,\dots,\mu_ k=0,\dots,n)\), \(\square =\eta^{\mu \nu}\partial^ 2/\partial x^{\mu}\partial x^{\nu}\), where \(\eta\) is the Minkowski metric in \(\mathbb R^{n+1}\), \(\eta =diag(-1,+1,\dots,+1)\); f(u,v,w) indicates the N functions \(f^ A(u,v,w)\), \((A=1,\dots,N)\) of class \(C^{\infty}\), defined in the domain \({\mathcal U}\times\mathbb R^{N(n+1)(n+2)/2}\); \({\mathcal U}\) is an open set of \(\mathbb R^ N\times \mathbb R^{N(n+1)}\) 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.


35L70 Second-order nonlinear hyperbolic equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
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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