Asymptotic behavior of solutions for wave equations in exterior domains

针对项目中出现的一些问题，笔者得出了几点结论。(1)当外域边界附近的耗散有效时，作者和广岛大学Ikehata教授（Hiroshima university）得到波动方程解的总能量和L^2范数随时间趋于无穷而衰减为0。对于整个空间的柯西问题，有川岛中尾一的著作（J.Math.Soc）。日本,1995)。他们将通常的能量方法与L^p-L^q估计相结合，讨论了能量衰减。与此方法相反，我们研究了L^2框架中L^2界的能量衰减。此外，我们的方法可以适用于导出R^3中可压缩Navier-Stokes方程密度的L^2衰减，这是由T.Kobayashi（九州理工学院）和R. ikehata完成的。(2)我们证明了当边界附近的耗散有效时，对于某些特定的初始数据，能量一般不会衰减。更直观地说，如果在捕获区域耗散有效，那么波就会在那里衰减。否则，波就会逃逸。为了证明这一点，我们采用了mochizuki教授提出的加权能量法，导出了局部能量时空积分的可积性，并利用其估计讨论了能量的非衰减性。此外，值得注意的是，这个估计可以适用于提高局部能量的衰减率，这是由m.b akao教授（j.f ff. eq .1998）推导出来的。(3)利用加权能量法，研究了R^3外星形域非线性耗散波方程经典解的存在性。此外，我们还证明了能量一般不会衰减。我们的关键思想是将非线性耗散项不仅视为耗散项，而且视为摄动项。特别地，当定义域是整个空间R^3时，我们推导出局部能量以一定的速率衰减到0。其速率可由耗散系数的衰减速率决定。利用R^3中自由波动方程的解的表示公式，证明了局部能量衰减。不幸的是，由于我们不知道自由波方程外域解的表示公式，所以在外域问题中局部能量是否衰减是开放的。作者在日本几所大学举办的偏微分方程研讨会上发表了这些结果。少

The author have obtained several results concerning with some problems arising in our project.(1) When the dissipation is effective near the boundary in an exterior domain, the author and Prof. Ikehata (Hiroshima Univ.) have obtained that the total energy and L^2 norm of solutions for the wave equations decay to 0 as time goes to infinity. For the Cauchy problem in the whole space there is a work of Kawashima-Nakao-One (J.Math.Soc.Japan, 1995). Combining the usual energy method with the L^p-L^q estimate, they discussed the energy decay. Contrary to this method, we investigated the energy decay or L^2 bound in L^2 framework. Further, our method can be applicable to derive the L^2 decay for the density for the compressible Navier-Stokes equations in R^3, which was done with T.Kobayashi (Kyushu Inst.Tech.) and R.Ikehata.(2) We have proved that when the dissipation is effective around the boundary, the energy does not in general decay for some specified initial data. Intuitively speaking, … More if the dissipation is effective in the trapping region, then the wave would decay there. Otherwise, the wave would escape. For the proof, we employed the weighted energy method due to Prof.K.Mochizuki to derive the integrability of space-time integral of the local energy and utilized its estimate to discuss the energy nondecay . Furthermore, it is noteworthy mentioning that this estimate can be applicable to improve the decay rate of the local energy, which was derived by Prof.M.Nakao (J.Diff.Eq.1998).(3) Employing the weighted energy method, the author investigated the existence of classical solutions for the nonlinear dissipative wave equations in an exterior star-shaped domain in R^3. Moreover, we proved also that the energy does not in general decay. Our crucial idea is to treat the nonlinar dissipative term not only as a dissipation but also a perturbation. In particular, when the domain is the whole space R^3, we derived that the local energy decays to 0 at a certain rate. Its rate can be determined by the decay rate of the dissipation coefficient. Utilizing the representing formula of solutions to the free wave equations in R^3, we have proved the local energy decay. Unfortunately, since we do not know the representing formula of solutions to the free wave equations in exterior domains, it is open whether the local energy decays or not in exterior problem. The author lectured these results in Semiar on Partial Differential Equations held at several Japanese Universities. Less