Wave Seminar Asymptotic behavior of the solutions to some Schr\"{o}dinger equations including subcritical dissipative nonlinearities

2007-07-13 16:00 - 2007-07-13 17:30
Faculty of Science Building #4 Room 409
Naoyasu Kita (University of Miyazaki)
This work is the collaboration with Akihiro Shimomura(Gakusyuin University). We consider the Cauchy problem of someSchr\"{o}dinger equations including subcritical dissipativenonlinearites, where the space dimension is limited to one.Our aim in this talk is to observe large-time asymptotic behaviors ofthe solutions, which was intensively studied by many mathematicians ifthe power of nonlinearity is critical (in other words, cubic) orsuper-critical. On the other hand, the subcritical case has not beenconsidered so frequently because the nonlinearity is believed to badlydecay. However, by imposing the dissipative structure on thenonlinearity, it is possible to clarify not only the decay rate of thesolution in the uniform norm but also its asymptotic form. Our argumentis largely based on Hayashi-Naumkin's method, but somewhat delicatetreatment of the nonlinear structure is required.