Some remarks on the local energy decay of solutions of the initial-boundary value problem for the wave equation in unbounded domains
Some remarks on the local energy decay of solutions of the initial-boundary value problem for the wave equation in unbounded domains
复制标题
关于无界域波动方程初边值问题解局部能量衰减的一些注记
DOI：
10.1016/0022-0396(77)90123-1
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发表时间：
1977
期刊：
影响因子：
--
通讯作者：
H. Walker
中科院分区：
文献类型：
--
作者：
H. Walker
假设&3是光滑的，在这种情况下（*）允许一个唯一解u (x, t)，使得对u (t) F={u， ut}在Z(9)中对于每一次t。实际上，在[4]中表明，算子u (t)是由S(9)上的一个斜自伴随算子a =(zi)生成的x(9)上的一组任意算子。Tt可以得出11u (t) F/IE （9j= I/F Ile(3)）对于所有FFX(9)和所有时间I，即1%溶液中超过9的总能量是守恒的。许多作者研究了（*）在紧子集R = 9上解的局部能量范数在大时间内的行为。Morawetz[5,61]表明，如果9是有限星形体的外部，则存在局部能量衰减的统一估计
It is assumed that &3 is smooth, in which case (*) admits a unique solution u (x, t) which is such that the pair U (t) F={u, ut} is in Z (9) for each time t. In fact, it is shown in [4] that the operators U (t) comprise the group of iuntary operators on X (9) generated by A=(zi), a skew-selfadjoint operator on S (9). Tt follows that 11 U (t) F/IE (9j= I/F Ile (3) for all FFX (9) and all time I, ie, the total energy over 9 of the solution of 1%) is conserved. A number of authors have studied the behavior for large time of the local energy norm of the solution of (*) over a compact subset R of 9. Morawetz [5, 61 showed that if 9 is the exterior of a finite star-shaped body, then there exists a uniform estimate of local energy decay