Geodesic
A sharp constant in a Sobolev--Nirenberg inequality and its application to the Schr\"odinger equation
Izvestiya. Mathematics , Tome 73 (2009) no. 3, pp. 555-577

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We prove that the solution of the Cauchy problem for a non-linear Schrödinger evolution equation with critical and supercritical exponents can blow up at a finite time for some initial data, and this time is estimated from above and below. To this end, an interpolation Nirenberg-type inequality and a Sobolev-type inequality are proved and the values of sharp constants in these inequalities are calculated.
Keywords: Nirenberg–Sobolev inequality, non-linear Schrödinger equation, blow-up, global solubility.
Mots-clés : sharp constant 
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     author = {Sh. M. Nasibov},
     title = {A sharp constant in a {Sobolev--Nirenberg} inequality and its application to the {Schr\"odinger} equation},
     journal = {Izvestiya. Mathematics },
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     publisher = {mathdoc},
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     year = {2009},
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     url = {http://geodesic.mathdoc.fr/item/IM2_2009_73_3_a4/}
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Sh. M. Nasibov. A sharp constant in a Sobolev--Nirenberg inequality and its application to the Schr\"odinger equation. Izvestiya. Mathematics , Tome 73 (2009) no. 3, pp. 555-577. http://geodesic.mathdoc.fr/item/IM2_2009_73_3_a4/