Minimal blow-up solutions to the mass-critical inhomogeneous NLS equation

Details Minimal blow-up solutions to the mass-critical inhomogeneous NLS equation Minimal blow-up solutions to the mass-critical inhomogeneous NLS equation

2009-04-08

arxiv.org/abs/0904.1317v2

Communications in Partial Differential Equations 36, 3 (2011) 487-531

Physics

Mathematical Physics

36 pages. More explanations, references updated, statement of Theorem 1.1 corrected. FInal version

Scientific paper

10.1080/03605302.2010.513410

We consider the mass-critical focusing nonlinear Schrodinger equation in the presence of an external potential, when the nonlinearity is inhomogeneous. We show that if the inhomogeneous factor in front of the nonlinearity is sufficiently flat at a critical point, then there exists a solution which blows up in finite time with the maximal (unstable) rate at this point. In the case where the critical point is a maximum, this solution has minimal mass among the blow-up solutions. As a corollary, we also obtain unstable blow-up solutions of the mass-critical Schrodinger equation on some surfaces. The proof is based on properties of the linearized operator around the ground state, and on a full use of the invariances of the equation with an homogeneous nonlinearity and no potential, via time-dependent modulations.

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