Uniform decay of local energy and the semi-linear wave equation on Schwarzschild space
Uniform decay of local energy and the semi-linear wave equation on Schwarzschild space
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DOI：
10.1007/s00220-006-0101-6
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发表时间：
2006-12-01
影响因子：
2.4
通讯作者：
Sterbenz, Jacob
中科院分区：
文献类型：
--
作者：
Blue, Pieter;Sterbenz, Jacob
我们给出了Schwarzschild背景下非齐次波动方程通解的局部能量的一致衰减估计。我们的估计表明，只要源项在范数(1-2m/r)(-1).(1+t+|r*|)L--1(1)(H-Omega(3)(r(2)dr*d omega))内有界，这类解具有渐近性质|Phi|=O(r(-1)\t-|r*|\(-1/2)).具体地说，这给出了对于所有2<sup>p</sup>的形式为Square(G)Phi=lambda|Phi|(P)Phi的非线性标量场的小幅度散射。
We provide a uniform decay estimate for the local energy of general solutions to the inhomogeneous wave equation on a Schwarzschild background. Our estimate implies that such solutions have asymptotic behavior |phi| = O(r(-1)\t - | r *|\(-1/2)) as long as the source term is bounded in the norm (1- 2M/r)(-1).(1+ t +| r *|)L--1(1)(H-Omega(3)(r(2)dr* d omega)). In particular this gives scattering at small amplitudes for non-linear scalar fields of the form square(g)phi =lambda|phi|(p)phi for all 2 < p.