Asymptotic behavior for wave equations with damping term

考虑当t趋于无穷时，具有耗散系数趋于0的抽象波动方程的初值问题。已知具有常数耗散项的波动方程的解是渐近自由的，如果耗散项的常数衰减且多项式阶小于-1。另一方面，在耗散系数为正常数的情况下，可知抽象波动方程的解与相应的抽象热方程的解之间的差衰减速度比每个解衰减得快（扩散现象）。本文给出了具有衰减耗散项的抽象波动方程解与相应的抽象抛物方程解之差的衰减估计。作为应用，我们得到了Dirichlet边界条件和Robin边界条件下耗散波动方程解的差值估计。其次，我们考虑了抽象的拟线性Kirchhoff型耗散双曲方程。已知全局解的唯一存在性是初始数据足够小。我们可以很容易地证明，当时间趋于无穷时，这个解趋向于相应的常系数热方程的解。然后考虑了带参数的耗散Kirchhoff方程趋向于相应的拟线性抛物方程的情况。对于每一个初始数据，已知如果耗散Kirchhoff方程与相应的抛物方程足够接近，则两个方程的唯一全局可解性和两个方程解之间的差值估计。然而，已知的估计是当地的时间。我们给出了结合渐近特性的全局衰减率估计。

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.