On study of partial differential equations describing natural phenomena

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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