Long Range Scattering for Nonlinear Schrödinger Equations 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.2010 Mathematics Subject Classification. 35B44, 35Q55, 35P25.

Long-range scattering for nonlinear Schrödinger equations with critical homogeneous nonlinearity in three space dimensions

“…We suppose that the nonlinearity F is homogeneous of degree 1 + 2/d, that is, F satisfies (1.1) F (λu) = λ 1+ 2 d F (u) for any u ∈ C and λ > 0. This is the continuation of the previous study in [10]. In [10], we consider one-and two-dimensional cases and give a sufficient condition on F : C → C for existence of a modified wave operator, that is, for that (NLS) admits a nontrivial solution which asymptotically behaves like…”

“…In [10], a sufficient condition on the nonlinearity F for existence of a modified wave operator is given in terms of the "Fourier coefficients" of the nonlinearity. The crucial step of construction of a modified wave operator is to find an asymptotic behavior that actually takes place.…”

“…For this part, specifying a resonant part of the nonlinearity, which determines the shape of the asymptotic behavior, is essential. A new ingredient in [10] is the expansion of the nonlinearity into an infinite sum via Fourier series expansion. For example, the nonlinearity F (u) = | Re u| Re u is written as…”

“…The case where the non-resonant part is a finite sum is previously treated in [7,8,16]. The main technical issue to treat general nonlinearity lies in showing that the non-resonant part which consists of infinitely many term is still acceptable (see [10]).…”

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In this paper, we consider the final state problem for the nonlinear Schrödinger equation with a homogeneous nonlinearity of the critical order which is not necessarily a polynomial. In [10], the first and the second authors consider one-and two-dimensional cases and gave a sufficient condition on the nonlinearity for that the corresponding equation admits a solution that behaves like a free solution with or without a logarithmic phase correction. The present paper is devoted to the study of the three-dimensional case, in which it is required that a solution converges to a given asymptotic profile in a faster rate than in the lower dimensional cases.

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