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On a global existence theorem of small amplitude solutions for nonlinear wave equations in an exterior domain. (English) Zbl 0592.35028
Let $$\Omega$$ be an unbounded domain in $${\mathbb R}^ n$$ with compact $$C^{\infty}$$ boundary $$\partial \Omega$$. Let $$t$$ or $$x_ 0$$ denote the time variable and $$x=(x_ 1,...,x_ n)$$ denote the space variables. Let $$\partial_ t=\partial /\partial t$$, $$\partial_ x=(\partial_ 1,...,\partial_ n)$$ with $$\partial_ j=\partial /\partial x_ j$$ $$(j=1,...,n)$$ and $$\underline{D}^ 1 D^ 1u=(\partial^ j_ t\partial_ x^{\alpha}u;\,1\leq j+| \alpha | \leq 2)$$. The mixed problem
$(\partial_ t^ 2-\Delta)u+F(t,x,\underline D^ 1D^ 1u)=f(t,x)\text{ in } [0,\infty)\times \Omega,$
$u=0\text{ on } [0,\infty)\times \partial \Omega,\tag{*}$
$u(0,x)=u_ 0(x), \quad (\partial_ tu)(0,x)=u_ 1(x) \text{ in }\Omega$ is considered. The authors prove that there exists a unique, global classical solution of (*) if the data $$u_ 0$$, $$u_ 1$$, and $$f$$ are small and sooth in some sense. They extend results due to J. Shatah [J. Differ. Equations 46, 409–425 (1982; Zbl 0518.35046)] to the mixed problem (*).
The proof is divided into two main steps. The first is to obtain an existence theorem for time-local solutions. As $$F$$ is fully nonlinear, one encounters loss of derivatives if the contraction mapping principle is used. One of the methods used to overcome this difficulty is the Nash-Moser technique. But for the regularity of solutions, such results seem somewhat rough. Using a method of P. A. Dionne [J. Anal. Math. 10, 1–90 (1962; Zbl 0112.32301)] one can overcome this difficulty for the Cauchy problem, but not for (*). However, by differentiating the differential equation in (*) with respect to $$t$$ and setting $$v=\partial_ t u$$ one obtains two equations for $$u$$ and $$v$$. The first, original equation is regarded as a fully nonlinear elliptic equation for $$u$$ and the second as a quasi-linear equation for $$v$$. The authors thus obtain an existence theorem via the usual contraction principle. In particular, if the data are sufficiently smooth, the time-local solution obtained has the same regularity as the data.
The second step is to obtain an a priori estimate for time-local solutions. For this a uniform decay estimate plays a crucial rôle. The authors develop a new method to obtain this estimate under the assumption that the scattering obstacle does not trap rays.
Reviewer: D. Kazarinoff

##### 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 35B45 A priori estimates in context of PDEs 35B65 Smoothness and regularity of solutions to PDEs
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