Nonexistence of scattering and modified scattering states for some nonlinear Schrödinger equation with critical homogeneous nonlinearity

Abstract

We consider large time behavior of solutions to the nonlinear Schrödinger equation with a homogeneous nonlinearity of the critical order which is not necessarily a polynomial. We treat the case in which the nonlinearity contains non-oscillating factor $|u{|}^{1+2/d}$. The case is excluded in our previous studies. It turns out that there are no solutions that behave like a free solution with or without logarithmic phase corrections. We also prove nonexistence of an asymptotic free solution in the case that the gauge invariant nonlinearity is dominant, and give a finite time blow-up result.

15 pages

References in corpus (3)

• On well-posedness of generalized Korteweg-de Vries equation in scale critical ^L^r space
• Long range scattering for nonlinear Schrödinger equations with critical homogeneous nonlinearity
• Modified scattering for the Klein-Gordon equation with the critical nonlinearity in three dimensions