Mathematics – Analysis of PDEs
Scientific paper
2008-06-10
Mathematics
Analysis of PDEs
Scientific paper
We study the focusing 3d cubic NLS equation with H^1 data at the mass-energy threshold, namely, when M[u_0]E[u_0] = M[Q]E[Q]. In earlier works of Holmer-Roudenko and Duyckaerts-Holmer-Roudenko, the behavior of solutions (i.e., scattering and blow up in finite time) is classified when M[u_0]E[u_0] < M[Q]E[Q]. In this paper, we first exhibit 3 special solutions: e^{it}Q and Q^+, Q^-; here Q is the ground state, and Q^+, Q^- exponentially approach the ground state solution in the positive time direction, meanwhile Q^+ having finite time blow up and Q^- scattering in the negative time direction. Secondly, we classify solutions at this threshold and obtain that up to \dot{H}^{1/2} symmetries, they behave exactly as the above three special solutions, or scatter and blow up in both time directions as the solutions below the mass-energy threshold. These results are obtained by studying the spectral properties of the linearized Schroedinger operator in this mass-supercritical case, establishing relevant modulational stability and careful analysis of the exponentially decaying solutions to the linearized equation.
Duyckaerts Thomas
Roudenko Svetlana
No associations
LandOfFree
Threshold solutions for the focusing 3d cubic Schroedinger equation does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Threshold solutions for the focusing 3d cubic Schroedinger equation, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Threshold solutions for the focusing 3d cubic Schroedinger equation will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-264327