This article is cited in 6 papers

A sharp constant in a Sobolev–Nirenberg inequality and its application to the Schrödinger equation

Sh. M. Nasibov

Institute of Applied Mathematics, Baku State University

Abstract: 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, sharp constant, non-linear Schrödinger equation, blow-up, global solubility.

UDC: 517.946.51.9

MSC: Primary 35J10; Secondary 35Q55, 46E35, 74H35

Received: 01.06.2007
Revised: 29.02.2008

DOI: 10.4213/im2671

 English version: Izvestiya: Mathematics, 2009, 73:3, 555–577

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