Geodesic
On the inhomogeneous nonlinear Schrödinger equation with harmonic potential and unbounded coefficient
Czechoslovak Mathematical Journal, Tome 60 (2010) no. 3, pp. 715-736.

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By deriving a variant of interpolation inequality, we obtain a sharp criterion for global existence and blow-up of solutions to the inhomogeneous nonlinear Schrödinger equation with harmonic potential $$ {\rm i}\varphi _t=-\triangle \varphi +|x|^2\varphi -|x|^b|\varphi |^{p-2}\varphi . $$ We also prove the existence of unstable standing-wave solutions via blow-up under certain conditions on the unbounded inhomogeneity and the power of nonlinearity, as well as the frequency of the wave.
Classification : 35J20, 35Q55
Keywords: interpolation inequality; inhomogeneous nonlinear Schrödinger equation; harmonic potential; blow-up; global existence; standing waves; strong instability 
@article{CMJ_2010__60_3_a9,
     author = {Chen, Jianqing},
     title = {On the inhomogeneous nonlinear {Schr\"odinger} equation with harmonic potential and unbounded coefficient},
     journal = {Czechoslovak Mathematical Journal},
     pages = {715--736},
     publisher = {mathdoc},
     volume = {60},
     number = {3},
     year = {2010},
     mrnumber = {2672412},
     zbl = {1224.35083},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2010__60_3_a9/}
}
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Chen, Jianqing. On the inhomogeneous nonlinear Schrödinger equation with harmonic potential and unbounded coefficient. Czechoslovak Mathematical Journal, Tome 60 (2010) no. 3, pp. 715-736. http://geodesic.mathdoc.fr/item/CMJ_2010__60_3_a9/