A sharp constant in a Sobolev--Nirenberg inequality and its application to the Schr\"odinger equation

Sh. M. Nasibov

Izvestiya. Mathematics , Tome 73 (2009) no. 3, pp. 555-577.

Résumé

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.

Détail

Keywords: Nirenberg–Sobolev inequality, non-linear Schrödinger equation, blow-up, global solubility.
Mots-clés : sharp constant

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