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Uniform decay estimates and the Lorentz invariance of the classical wave equation. (English) Zbl 0635.35059
The study of the problem of global existence of solutions to general classes of nonlinear wave equations subject to “small” initial conditions makes use as essential ingredient of the uniform decay property of solutions to the classical wave equation. That is, let u be the solution of the problem \(\square u=u_{tt}-\Delta u=0\), \(u=0\), \(u_ t=g(x)\) at \(t=0\), with g a smooth, compactly supported function in \({\mathbb{R}}^ n.\) Then [see the author, Commun. Pure Appl. Math. 33, 43-101 (1980; Zbl 0405.35056)] \[ (*)\quad | u(t,x)| \leq ct^{-(n-1)/2} \| g\|_{W^{[n/2],1}} \] for all \(x\in {\mathbb{R}}^ n,\) \(t>0\) and \(W^{s,1}\), \(s\in N\) in the classical Sobolev spaces in \({\mathbb{R}}^ n.\) The estimate (*) contains the L 1 norms of derivatives of g which is a drawback when applied to nonlinear equations.
The purpose of the paper is to replace the \(W^{[n/2],1}\) norm in (*) by some \(W^{s,2}\)-norm. The author accomplishes this by a suitable modification of Sobolev spaces \(W^{s,2}\) and applies it to the nonlinear wave equation \(\square u=F(u',u'')\), with the initial data \(u=\epsilon f(x)\), \(u_ t=\epsilon g(x)\) at \(t=0\). F is a smooth function of (u’,u”), the first and second space time derivatives of u(t,x), \(x\in {\mathbb{R}}^ n,\) vanishing together with its first derivatives for \((u',u'')=0\), and f,g \(C_ 0^{\infty}\) and \(\epsilon >0\) a small parameter.
The main result is that for \(n>3\), all solutions with sufficiently small initial data remain smooth for all time, whereas for \(n=3\), there exist constants \(a_ 0\), A such that for any \(0<a\leq a_ 0\), \(T\geq \exp [A/a]\), where T is the life span defined as \(T=\sup \beta >0\) for which \(C^{\infty}\)-solution exists for all \(x\in {\mathbb{R}}^ n,\) \(0\leq t<\beta\).
Reviewer: E.Young

MSC:
35L70 Second-order nonlinear hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
35L15 Initial value problems for second-order hyperbolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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