On study of partial differential equations describing natural phenomena

1，P.I.Naumkin和我研究了具抽运的Burgers方程，利用适当的变换和非线性项的结构，证明了了解的时间存在性和解的渐近性。2，E.I.Kaiina和我研究了原点边值为0的半直线上的KdV方程。AIR函数在左手边快速振荡，在右手边呈指数衰减。利用这一性质，我们证明了KdV方程解的渐近性。3，E.I.Kaiina，P.I.Naumkin和我研究了具有亚临界非线性的非线性复耗散方程，证明了解在自相似解的邻域内是稳定的。4，P.I.Naumkin，Shimomura，Tonegawa和我共同研究了具有立方非线性的非线性Schredinger方程。已知在一定的几何假设下，最终数据存在修正的波动算子。通过找到一种新的方法来获得问题的二次近似解，我们成功地去掉了一个强几何假设。5，E.I.Kaiina，P.I.Naumkin和我研究了具有超临界或临界非线性的非线性阻尼波方程。在前人的工作中，已知一个整体存在定理在空间维小于5。我们利用加权Soblev空间和线性问题解的估计改进了这一结果。此外，在临界情况下，我们证明了解的渐近性。结果表明，解的时间衰减阶数高于线性问题解的时间衰减阶。我们用研究非线性耗散方程的方法得到了结果。

1,P.I.Naumkin and I studied the Burgers equation with pumping and showed a existence in time of solutions and asymptotic behavior of solutions by using a suitable transformation and the structure of nonlinear term.2,E.I.Kaikina and I studied the KdV equations in a half line with 0 boundary value at the origin. Airy function is oscillating rapidly in the left hand side and decaying exponentially in the right hand side. We showed asymptotics of solutions to the KdV equation by making use of this property.3,E.I.Kaikina, P.I.Naumkin and I studied nonlinear complex dissipative equations with sub-critical nonlinearities and showed a solution is stable in the neighborhood of a self similar, solution.4,P.I.Naumkin, Shimomura, Tonegawa and I did a joint work on nonlinear Schredinger equations with cubic nonlinearities. It was known that there exists a modified wave operator under some geometric assumptions on the final data. We succeeded to remove a strong geometric assumption by finding a new way to get a second approximate solution of the problem.5,E.I.Kaikina, P.I.Naumkin and I studied nonlinear damped wave equations with super-critical or critical nonlinearities. In the previous works, it was known that a global existence theorem holds in space dimension is less than 5. We improved this result for any space dimension by using the weighted Sobolev spaces and estimates of solutions linear problem. Furthermore, in the critical case we showed asymptotics of solutions. The result implies the decay order in time of solutions is higher than that of solutions to linear problem. We obtained the results by using the method we found in the study of nonlinear dissipative equations