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arXiv · 1706.03491

Long range scattering for nonlinear Schr\"odinger equations with critical homogeneous nonlinearity in three space dimensions

Satoshi Masaki·Hayato Miyazaki·Kota Uriya
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Abstract

In this paper, we consider the final state problem for the nonlinear Schr\"odinger 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.

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Satoshi Masaki, Hayato Miyazaki, Kota Uriya. 2017-06-12. Long range scattering for nonlinear Schr\"odinger equations with critical homogeneous nonlinearity in three space dimensions. https://doi.org/10.1090/tran/7636

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