Journal
SIAM JOURNAL ON MATHEMATICAL ANALYSIS
Volume 54, Issue 1, Pages 173-219Publisher
SIAM PUBLICATIONS
DOI: 10.1137/21M1406003
Keywords
Schrequation; inverse square potential; energy critical; stable manifold
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Funding
- Jiangsu Shuang Chuang Doctoral Plan
- NSF of Jiangsu (China) [BK20200346, BK20190323]
- National Science Foundation [DMS-1900083]
- Simons Foundation
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This article examines the focusing energy critical nonlinear Schrodinger equation with inverse square potential in dimensions 3, 4, and 5. The characteristics of solutions on the energy surface of the ground state are described and proved. It is shown that solutions with kinetic energy less than that of the ground state must converge to the ground state, while solutions with greater kinetic energy will blow up in finite time.
We consider the focusing energy critical nonlinear Schrodinger equation (NLS) with inverse square potential in dimension d = 3, 4, 5 with the details given in d = 3 and remarks on results in other dimensions. Solutions on an energy surface of the ground state are characterized. We prove that solutions with kinetic energy less than that of the ground state must scatter to zero or belong to the stable/unstable manifolds of the ground state. In the latter case they converge to the ground state exponentially in the energy space as t -> infinity or t -> -infinity. (In three-dimensions without radial assumption, this holds under the compactness assumption of nonscattering solutions on the energy surface.) When the kinetic energy is greater than that of the ground state, we show that all radial H-1 solutions blow up in finite time, with the only two exceptions being in the case of five-dimensions which belong to the stable/unstable manifold of the ground state. The proof relies on detailed spectral analysis, local invariant manifold theory, and a global Virial analysis.
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