#### DOI

10.17077/etd.88zis0xk

#### Document Type

Dissertation

#### Date of Degree

Spring 2017

#### Access Restrictions

.

#### Degree Name

PhD (Doctor of Philosophy)

#### Degree In

Applied Mathematical and Computational Sciences

#### First Advisor

Zhang, Xiaoyi

#### First Committee Member

Zhang, Xiaoyi

#### Second Committee Member

Li, Tong

#### Third Committee Member

Jorgensen, Palle

#### Fourth Committee Member

Strohmer, Gerhard

#### Fifth Committee Member

Wang, Lihe

#### Abstract

We consider the Cauchy problem for the focusing energy critical NLS with inverse square potential. The energy of the solution, which consists of the kinetic energy and potential energy, is conserved for all time. Due to the focusing nature, solution with arbitrary energy may exhibit various behaviors: it could exist globally and scatter like a free evolution, persist like a solitary wave, blow up at finite time, or even have mixed behaviors. Our goal in this thesis is to fully characterize the solution when the energy is below or at the level of the energy of the ground state solution $W_a$. Our main result contains two parts.

First, we prove that when the energy and kinetic energy of the initial data are less than those of the ground state solution, the solution exists globally and scatters.

Second, we show a rigidity result at the level of ground state solution. We prove that among all solutions with the same energy as the ground state solution, there are only two (up to symmetries) solutions $W_a^+, W_a^-$ that are exponential close to $W_a$ and serve as the threshold of scattering and blow-up. All solutions with the same energy will blow up both forward and backward in time if they go beyond the upper threshold $W_a^+$; all solutions with the same energy will scatter both forward and backward in time if they fall below the lower threshold $W_a^-$.

In the case of NLS with no potential, this type of results was first obtained by Kenig-Merle \cite{R: Kenig focusing} and Duyckaerts-Merle \cite{R: D Merle}. However, as the potential has the same scaling as $\Delta$, one can not expect to extend their results in a simple perturbative way. We develop crucial spectral estimates for the operator $-\Delta+a/|x|^2$, we also rely heavily on the recent understanding of the operator $-\Delta+a/|x|^2$ in \cite{R: Harmonic inverse KMVZZ}.

#### Public Abstract

Nonlinear Schrödinger equation(NLS) with singular potential has applications in quantum mechanics and nonlinear optics. In this thesis, we study the focusing energy critical NLS with inverse square potential. The energy of the solution, which consists of kinetic energy and potential energy, is conserved for all time. The focusing nonlinearity represents the attractive force among all microscopic particles. Depending on the competition between the scattering effect from the linear term and the focusing effect from the nonlinearity, a solution with arbitrary energy may exhibit various behaviors: it could exist globally and scatter like a free evolution, persist like a solitary wave, blow up at finite time, or have mixed behaviors. We fully characterize the solution when the energy of the solution is below or at the level of the energy of the ground state solution *W*.

Our main result contains two parts. First, we prove that when the energy and kinetic energy of the initial data are less than those of the ground state solution, the solution exists globally and scatters. Second, we classify the dynamical behavior of the solution assuming that the energy of the solution equals the energy of *W*. Specifically, if the kinetic energy of the solution is greater than that of *W*, the solution either blows up at finite time or has mixed behaviors; if the kinetic energy of the solution is less, the solution either scatters or has mixed behaviors, if both kinetic energies are the same, the solution persists like a solitary wave.

#### Keywords

blow-up, dynamics, ground state solution, inverse square potential, NLS, scattering

#### Pages

xi, 105 pages

#### Bibliography

Includes bibliographical references (pages 101-105).

#### Copyright

Copyright © 2017 Kai Yang

#### Recommended Citation

Yang, Kai. "Dynamics of the energy critical nonlinear Schrödinger equation with inverse square potential." PhD (Doctor of Philosophy) thesis, University of Iowa, 2017.

https://doi.org/10.17077/etd.88zis0xk