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- Title
FINITE-TIME BLOWUP FOR A SCHRÖDINGER EQUATION WITH NONLINEAR SOURCE TERM.
- Authors
Cazenave, Thierry; Martel, Yvan; Zhao, Lifeng
- Abstract
We consider the nonlinear Schrödinger equation ut = iΔu + |u|αu on ℝN, α > 0, for H¹-subcritical or critical nonlinearities: (N - 2) α ≤ 4. Under the additional technical assumptions a ≥ 2 (and thus N ≤ 4), we construct H¹ solutions that blow up in finite time with explicit blow-up profiles and blow-up rates. In particular, blowup can occur at any given finite set of points of ℝN. The construction involves explicit functions U, solutions of the ordinary differential equation Ut = |U|αU. In the simplest case, U(t, x) = (|x|k - αt)-1/α for t < 0, x ∈ ℝN. For k sufficiently large, U satisfies |ΔU| ≪ Ut close to the blow-up point (t, x) = (0, 0), so that it is a suitable approximate solution of the problem. To construct an actual solution u close to U, we use energy estimates and a compactness argument.
- Subjects
NONLINEAR theories; SCHRODINGER equation
- Publication
Discrete & Continuous Dynamical Systems: Series A, 2019, Vol 39, Issue 2, p1171
- ISSN
1078-0947
- Publication type
Article
- DOI
10.3934/dcds.2019050