On a global existence theorem of small amplitude solutions for nonlinear wave equations in an exterior domain.

*(English)*Zbl 0592.35028Let \(\Omega\) 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.

\[ (\partial_ t^ 2-\Delta)u+F(t,x,\underline D^ 1D^ 1u)=f(t,x)\text{ in } [0,\infty)\times \Omega, \]

\[ u=0\text{ on } [0,\infty)\times \partial \Omega,\tag{*} \]

\[ u(0,x)=u_ 0(x), \quad (\partial_ tu)(0,x)=u_ 1(x) \text{ in }\Omega \] 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 |

##### Keywords:

mixed problem; unique, global classical solution; existence theorem; Nash-Moser technique; regularity of solutions; contraction principle; uniform decay estimate; scattering obstacle
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\textit{Y. Shibata} and \textit{Y. Tsutsumi}, Math. Z. 191, 165--199 (1986; Zbl 0592.35028)

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