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arXiv · 1505.06496

Well-posedness and scattering for fourth order nonlinear Schr\"odinger type equations at the scaling critical regularity

Hiroyuki Hirayama·Mamoru Okamoto
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Abstract

In the present paper, we consider the Cauchy problem of fourth order nonlinear Schr\"odinger type equations with a derivative nonlinearity. In one dimensional case, we prove that the fourth order nonlinear Schr\"odinger equation with the derivative quartic nonlinearity $\partial _x (\overline{u}^4)$ is the small data global in time well-posed and scattering to a free solution. Furthermore, we show that the same result holds for the $d \ge 2$ and derivative polynomial type nonlinearity, for example $|\nabla | (u^m)$ with $(m-1)d \ge 4$.

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Hiroyuki Hirayama, Mamoru Okamoto. 2015-05-24. Well-posedness and scattering for fourth order nonlinear Schr\"odinger type equations at the scaling critical regularity. https://doi.org/10.3934/cpaa.2016.15.831

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