Asymptotic behavior and singular limit problem for dissipative hyperbolic equations

我们考虑了带耗散项的耗散Kirchhoff方程，其中耗散项系数随时间的变化而衰减。首先，我们考虑耗散的Kirchhoff方程，其中耗散项的系数依赖于时间和空间价值，并且相对于时间价值的衰减慢于临界指数。然后证明了具有小Sobolev范数的初值问题解的唯一整体存在性。其次，我们考虑了耗散的Kirchhoff方程，其中耗散项的系数只依赖于迅速衰减的时间价值。然后证明了这类函数的唯一整体可解性和散射解的存在性。

We considered dissipative Kirchhoff equation with dissipation, where the coefficient of the dissipation decays with respect to the time valuable. First we consider dissipative Kirchhoff equation where the coefficient of the dissipation term depends time and space valuables and decays slowly than the critical exponent with respect to the time valuable. Then we proved the unique global existence of the solution for initial data with small Sobolev norm. Secondly we considered the dissipative Kirchhoff equation where the coefficient of the dissipation term depends only on time valuable which decays rapidly. Then we showed the unique global solvability and existence of the scattering on some class of the functions.