Sharp upper and lower bounds on the blow-up rate for nonlinear Schrödinger equation with potential

2007 ◽  
Vol 190 (2) ◽  
pp. 1267-1272
Author(s):  
Shihui Zhu ◽  
Xiaoguang Li
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Lower Bounds ◽  
Schrodinger Equation ◽  
Blow Up ◽  
Nonlinear Schrodinger Equation ◽  
Upper And Lower Bounds ◽  
Nonlinear Schrödinger ◽  
Blow Up Rate

scholarly journals Sharp blowup rate for NLS with a repulsive harmonic potential

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Rui Zhou
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Lower Bounds ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Harmonic Potential ◽  
Upper And Lower Bounds ◽  
Singular Solutions ◽  
Blowup Rate ◽  
Nonlinear Schrödinger

AbstractIn this paper, we are concerned with the blowup solutions of the $L^{2}$ L 2 critical nonlinear Schrödinger equation with a repulsive harmonic potential. By using the results recently obtained by Merle and Raphaël and by Carles’ transform we establish in a quite elementary way universal and sharp upper and lower bounds of the blowup rate for the blowup solutions of the aforementioned equation. As an application, we derive upper and lower bounds on the $L^{r}$ L r -norms of the singular solutions.

scholarly journals The structure of solutions to the pseudo-conformally invariant nonlinear Schrödinger equation

1991 ◽  
Vol 117 (3-4) ◽  
pp. 251-273 ◽  
Author(s):  
Thierry Cazenave ◽  
Fred B. Weissler
Keyword(s):  
Schrödinger Equation ◽  
Global Solution ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Blow Up ◽  
Global Solutions ◽  
Nonlinear Schrodinger Equation ◽  
Structure Of Solutions ◽  
Nonlinear Schrödinger ◽  
Conformally Invariant

SynopsisWe study solutions in ℝn of the nonlinear Schrödinger equation iut + Δu = λ |u|γu, where γ is the fixed power 4/n. For this particular power, these solutions satisfy the “pseudo-conformal” conservation law, and the set of solutions is invariant under a related transformation. This transformation gives a correspondence between global and non-global solutions (if λ < 0), and therefore allows us to deduce properties of global solutions from properties of non-global solutions, and vice versa. In particular, we show that a global solution is stable if and only if it decays at the same rate as a solution to the linear problem (with λ = 0). Also, we obtain an explicit formula for the inverse of the wave operator; and we give a sufficient condition (if λ < 0) that the blow up time of a non-global solution is a continuous function on the set of initial values with (for example) negative energy.

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