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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 $$u^ A$$ $$(A=1,\dots,N)$$, $$\partial^ ku=\partial^ ku^ A/\partial x^{\mu_ 1}\dots\partial^{\mu_ k}$$ $$(\mu_ 1,\dots,\mu_ k=0,\dots,n)$$, $$\square =\eta^{\mu \nu}\partial^ 2/\partial x^{\mu}\partial x^{\nu}$$, 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.

##### MSC:
 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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##### References:
  Christodoulou, C. R. Acad. Sci Paris 293 pp 39– (1981)  and , Existence of global solutions of the Yang-Mills, Higgs and Spinor field equations in 3 + 1 dimensions, Annales des Ecole Normale Superieur, 4th Series, 14, 1981, pp. 481–506. · Zbl 0499.35076  Klainerman, Comm. Pure and Appl. Math. 33 pp 43– (1980)  Long time behavior of solutions to nonlinear wave equations, Proceedings of the International Congress of Mathematicians, Warsaw, 1982.  Conformal treatment of infinity, in Relativity, Groups and Topology, and , (eds.), Gordon and Breach, 1963.
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