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Normal forms and quadratic nonlinear Klein-Gordon equations. (English) Zbl 0597.35101
This is a companion paper to the one of S. Klainerman [Commun. Pure Appl. Math. 38, 631-641 (1985; see the preceding review)]. As in that paper it is shown that a global smooth solution of \[ (\partial^ 2_ t-\Delta +1)u+f(u,\partial u,\partial^ 2u)=0;\quad u(0)=u_ 0,\quad \partial_ tu(0)=u_ 1 \] exists, if the initial values are small, and if \(f(u,\partial u,\partial^ 2u)\) vanishes of second order at \((u,\partial u,\partial^ 2u)=0\). Here \(u=u(x,t)\) with \(x\in {\mathbb{R}}^ n\) and \(n>2\). A different method of proof is used.
The problem is transformed to a similar problem with a function f vanishing of third order at 0, and decay estimates are proved for the new problem. The Sobolev norm of the solution obtained decays like \(t^{- n/2}\).
Reviewer: H.D.Alber

MSC:
35Q99 Partial differential equations of mathematical physics and other areas of application
35L05 Wave equation
35B40 Asymptotic behavior of solutions to PDEs
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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References:
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