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Global existence of small radially symmetric solutions to quadratic nonlinear wave equations in an exterior domain

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Abstract

We study the initial boundary value problem for the nonlinear wave equation:

$$\left\{ \begin{gathered} \partial _t^2 u - (\partial _r^2 + \frac{{n - 1}}{r}\partial _r )u = F(\partial _t u,\partial _t^2 u),t \in \mathbb{R}^ + ,R< r< \infty , \hfill \\ u(0,r) = \in _0 u_0 (r),\partial _t u(0,r) = \in _0 u_1 (r),R< r< \infty , \hfill \\ u(t,R) = 0,t \in \mathbb{R}^ + , \hfill \\ \end{gathered} \right.$$
((*))

wheren=4,5,u0,u1 are real valued functions and ∈0 is a sufficiently small positive constant. In this paper we shall show small solutions to (*) exist globally in time under the condition that the nonlinear termF:ℝ2→ℝ is quadratic with respect to ∂ t u and ∂ 2 t u.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Engineering, Gunma University, 376, Kiryu, Japan

    Nakao Hayashi

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  1. Nakao Hayashi
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Hayashi, N. Global existence of small radially symmetric solutions to quadratic nonlinear wave equations in an exterior domain. Manuscripta Math 81, 15–39 (1993). https://doi.org/10.1007/BF02567842

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  • DOI: https://doi.org/10.1007/BF02567842

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