Zeitschrift für Analysis und ihre Anwendungen
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An Inhomogeneous, $L^2$-Critical, Nonlinear Schrödinger EquationFrançois Genoud (1) Dept of Mathematics and The Maxwell Institute of Mathematical Sciences, Heriot-Watt University, EH14 4AS, Edinburgh, UK
An inhomogeneous nonlinear Schrödinger equation is considered, which is invariant under $L^2$-scaling. The sharp condition for global existence of $H^1$-solutions is established, involving the$L^2$-norm of the ground state of the stationary equation. Strong instability of standing waves is proved by constructing self-similar solutions blowing up in finite time.
Keywords: Global existence, blow-up, $L^2$-critical, inhomogeneous NLS
Genoud François: An Inhomogeneous, $L^2$-Critical, Nonlinear Schrödinger Equation. Z. Anal. Anwend. 31 (2012), 283-290. doi: 10.4171/ZAA/1460