Geodesic
Gradient blow-up of solutions to the Cauchy problem for the Schr\"odinger equation
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function theory and equations of mathematical physics, Tome 283 (2013), pp. 171-187

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A class of nonlinear Schrödinger operators with singular coefficients is studied. For this class, necessary and sufficient conditions are established for the existence of initial data such that the corresponding solution to the Cauchy problem blows up in finite time. A regularization procedure for the Cauchy problem is proposed, and the limit behavior of the sequence of solutions to the regularized problems is analyzed. 
@article{TM_2013_283_a11,
     author = {V. Zh. Sakbaev},
     title = {Gradient blow-up of solutions to the {Cauchy} problem for the {Schr\"odinger} equation},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {171--187},
     publisher = {mathdoc},
     volume = {283},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2013_283_a11/}
}
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V. Zh. Sakbaev. Gradient blow-up of solutions to the Cauchy problem for the Schr\"odinger equation. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function theory and equations of mathematical physics, Tome 283 (2013), pp. 171-187. http://geodesic.mathdoc.fr/item/TM_2013_283_a11/