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

Abstract

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. To obtain the necessary convergence rate, we employ the end-point Strichartz estimate and modify a time-dependent regularizing operator, introduced in [10]. Moreover, we present a candidate of the second asymptotic profile to the solution.2010 Mathematics Subject Classification. 35B44, 35Q55, 35P25.

Modified scattering for the Klein-Gordon equation with the critical nonlinearity in three dimensions

“…Proof. We follow the argument in [22]. It is easy to see that c n = 0 for odd n and This completes the proof.…”

Large-time behavior of compressible polytropic fluids and nonlinear Schr{ö}dinger equation

“…In the case of (1.27), the long range effects are understood only in the critical case σ = 1/d: see [15] and references therein. See also [18] and references therein for the existence of wave operators (Cauchy problem with prescribed behavior at t = ∞ instead of t = 0) in the case σ = 1/d. It seems that so far, the long range scattering has not been studied for (1.27) in the case 0 < σ < 1/d.…”

Long Range Scattering for Nonlinear Schrödinger Equations with Critical Homogeneous Nonlinearity

“…More precisely, the behavior of a solution depends on the shape of the nonlinearity [3,7,8,15,16,18]. In [11,12], we introduce a decomposition of the nonlinearity (1.2) F (u) = g 0 |u| 1+ 2 d + g 1 |u|…”

“…According to these facts, we do not try to give a behavior in terms of {g n } n in this paper, but instead deny the existence of a solution that behaves like a free solution or a free solution with a logarithmic phase correction, that is, behaves like (1.4). This is a complementary study of [11,12], and is an extension of [17,19].…”

“…Indeed, g(θ) = | cos θ| 1/2 is such an example. The case corresponds to F (u) = | Re u| 1 2 |u| 1 2 + 2 d , and g n = O(|n| −3/2 ) (see [12,14]).…”