Asymptotic behavior of solutions for wave equations in exterior domains

对于本课题中出现的一些问题，作者得到了几个结果：(1)当边界附近的耗散是有效的时，作者和广岛大学的池方教授(Hiroshima University.)得到了波动方程的解的总能量和L^2范数随时间衰减为0。对于整个空间中的柯西问题，有一个Kawashima-Nakao-One的工作(J.Math.Soc.Japan，1995)。结合通常的能量方法和L^p-L^q估计，讨论了能量衰变。与这种方法相反，我们在L^2框架下研究了能量衰变或L^2束缚。此外，我们的方法也可以用来计算R^3中可压缩N-S方程密度的L^2衰变。(2)证明了当边界附近的耗散是有效的时，能量一般不会对某些特定的初值衰减。直观地说，…更重要的是，如果耗散在捕获区是有效的，那么波就会在那里衰减。否则，海浪就会逃逸。作为证明，我们利用K.Mochizuki教授的加权能量方法导出了局域能量的时空积分的可积性，并利用它的估计讨论了能量的非衰变。此外，值得一提的是，这个估计可以用来改善M.Nakao教授(J.Diff.Eq.1998)导出的局部能量的衰减率。(3)利用加权能量方法，研究了R^3中外部星形区域上的非线性耗散波动方程经典解的存在性。此外，我们还证明了能量一般不衰减。我们的关键思想是不仅将非线性耗散项视为耗散项，而且将其视为微扰。特别地，当区域是整个空间R^3时，我们推导出局域能量以一定的速率衰减到0。其衰减率可由耗散系数的衰减率来确定。利用R^3中自由波方程解的表示式，证明了局部能量衰减。不幸的是，由于我们不知道自由波动方程在外部区域的解的表示式，所以外部问题中的局部能量是否衰减是一个未知数。作者在日本几所大学举办的《偏微分方程论》中讲授了这些结果。较少

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