Stabilization problem for nonlinear wave eq

首席调查员。中尾。考虑了非线性波动方程在内部和外部区域的镇定问题，以及非线性热方程解的性态。我们首先证明了线性波动方程的局部能量衰减。并在此基础上得到了L^p的估计。再远一点。利用这些估计，我们讨论了半线性波动方程的整体存在性。我们注意到，在我们的论证中，对于内部问题，没有关于障碍物形状的几何条件。证明了具有很弱耗散项的拟线性波动方程整体光滑解的存在性。在这一过程中，我们证明了变系数波动方程的唯一延拓性质。中尾武彦的不等式被用于衰变估计。这是本文的创新之处。结合非线性热方程，我们处理了在各种非线性扰动下的平均曲率型拟线性方程和m-Laples型拟线性方程。我们得到了解的精确估计，包括渐近性、t→∞和t=0附近的光滑化效应。川岛研究员主要研究了与气体动力学有关的方程，并得到了许多有趣的结果。柴田研究员考虑了粘弹性波动方程和与流体力学有关的外部问题，并用谱分析证明了许多新的结果。加藤研究员讨论了一个非牛顿流动方程的整体解。

The head investigator. Nakao. has considered the stabilization problem for the nonlinear wave equations in interior and exterior domains and also the behaviours of solutions for nonlinear heat equations.For exterior problems. we first proved the local energy decay for linear wave equations. and on the basis of this we have derived L^p estimates. Further. by use of these estimates we have discussed on the global existence of semilinear wave equations. We note that in our argument no geometrical conditions on the shape of obstacles.For interior problems. we have proved global existence of smooth solutions for the quasilinear wave equations with a very weak dissipative term. In this procedure we have showed a unique continuation property for the wave equation with a variable coefficient. Nakao's inequality was used for the decay estimate. which is an originality of this paper.Concernibg nonlinear heat equations we have treated meancurvature type and m-laplacian type quasilinear equations under various nonlinear perturbations. We have derived sharp estimates of solutions including asymptotics as t → ∞ and smoothing effects near t = O.Investigator Kawashima has mainly treated the equations concerning gas dynamics and shown many interesting results. Investigator Shibata has considered the visco-elastic wave equations and also exterior problems concerning fluid dynamicsand prove many new results by use of spectral analysis. Investigator Kato has discussed on the global solutions of a non-Newtonian flow equation.