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Asymptotic behavior for wave equations with damping term

带阻尼项的波动方程的渐近行为

基本信息

批准号：
17540173
负责人：
YAMAZAKI Taeko
金额：
$ 0.83万
依托单位：
Tokyo University of Science
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2005
资助国家：
日本
起止时间：
2005 至 2006
项目状态：
已结题

项目摘要

We considered the initial value problem of the abstract wave equation with dissipation whose coefficient tends to 0 as t tends to infinity. It is known that the solution of the wave equation with a constant the dissipative term is asymptotically free if the constant of the dissipation decays with the polynomial order less than-1. On the other hand, in the case that the coefficient of the dissipation is a positive constant, it is known that the difference between the solution of the abstract wave equation and the solution of the corresponding abstract heat equation decays faster than each of the solution does (diffusion phenomenon). In this research, we showed the decay estimate of the difference between the solution of the abstract wave equation with decaying dissipative term and the solution of the corresponding abstract parabolic equation. As applications, we obtained the estimate of the difference between the solution of the dissipative wave equation with Dirichlet and Robin boundary conditions.Next, we considered the abstract quasilinear dissipative hyperbolic equation of Kirchhoff type. The unique existence of the global solution was known for sufficiently small initial data. We can easily show that this solution tends to the solution of the corresponding heat equation with a constant coefficient as the time tends to infinity. Then we considered the case that the dissipative Kirchhoff equation with parameter tends to the corresponding quasilinear parabolic equation. For every initial data, it is known that if the dissipative Kirchhoff equation is sufficiently close to the corresponding parabolic equation, the unique global solvability of the both equations and the estimate of difference between the solutions of the two equations. However the known estimates are local in time. We showed the global decay rate estimate combined with the asymptotic behavior.

考虑了当t趋于无穷大时系数趋于0的抽象耗散波动方程的初值问题。已知当耗散项为常数时，如果耗散项的常数以小于-1的多项式阶衰减，则波动方程的解是渐近自由的。另一方面，在耗散系数为正常数的情况下，已知抽象波动方程的解与对应的抽象热方程的解之间的差比每个解衰减得更快（扩散现象）。在本研究中，我们给出了具有衰减耗散项的抽象波动方程的解与相应的抽象抛物方程的解之差的衰减估计。作为应用，我们得到了耗散波动方程在Dirichlet和Robin边界条件下的解的差的估计。其次，我们考虑了抽象拟线性耗散双曲型Kirchhoff方程。对于足够小的初始数据，已知全局解的唯一存在性。我们可以很容易地证明，当时间趋于无穷大时，这个解趋于相应的常系数热方程的解。然后考虑了带参数的耗散Kirchhoff方程趋于相应的拟线性抛物方程的情形。对于每一个初始值，如果耗散的Kirchhoff方程与相应的抛物方程足够接近，那么这两个方程的唯一整体可解性和两个方程解之间的差的估计是已知的.然而，已知的估计在时间上是局部的。我们给出了全局衰减率估计和渐近行为。