Mathematische Annalen

, Volume 320, Issue 1, pp 11–31 | Cite as

$L^p$ estimates for the linear wave equation and global existence for semilinear wave equations in exterior domains

  • Mitsuhiro Nakao
Original articles


We consider the initial-boundary value problem for the semilinear wave equation



where \(\Omega\) is an exterior domain in \(R^N\), \(a(x)u_t\) is a dissipative term which is effective only near the 'critical part' of the boundary. We first give some \( L^p\) estimates for the linear equation by combining the results of the local energy decay and \(L^p\) estimates for the Cauchy problem in the whole space. Next, on the basis of these estimates we prove global existence of small amplitude solutions for semilinear equations when \(\Omega\) is odd dimensional domain . When \(N=3 \mbox{ and } f=|u|^\alpha u \) our result is applied if \(\alpha > 2\sqrt{3}-1\). We note that no geometrical condition on the boundary \(\partial \Omega\) is imposed.


Linear Equation Wave Equation Cauchy Problem Small Amplitude Global Existence 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Mitsuhiro Nakao
    • 1
  1. 1.Graduate School of Mathematics, Kyushu University, Ropponmatsu, Fukuoka 810–8560, Japan JP

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