Global existence of small radially symmetric solutions to quadratic nonlinear wave equations in an exterior domain |
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Authors: | Nakao Hayashi |
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Institution: | 1. Department of Mathematics, Faculty of Engineering, Gunma University, 376, Kiryu, Japan
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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,u 0,u 1 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:?2→? is quadratic with respect to ? t u and ? t 2 u. |
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