Blow-up in several points for the nonlinear Schrödinger equation on a bounded domain. (English) Zbl 1249.35303

Godet, Nicolas

Differ. Integral Equ. 24, No. 5-6, 505-517 (2011).

We consider the \(L^2\)-critical focusing nonlinear Schrödinger equation posed on a bounded and regular domain \(\Omega \) of \(\mathbb {R}^d\) (with \(d=2,3\)): \(i\partial _tu+\Delta u=-| u| ^{4/d}u\), \((t,x)\in [0,T)\times \Omega \) with initial data and the Dirichlet boundary condition.
In [Commun. Math. Phys. 129, No. 2, 223–240 (1990; Zbl 0707.35021)] F. Merle shows that if \(\Omega \) is the whole space \(\mathbb {R}^d\) (without restriction on \(d\)) then given \(p\) points in \(\mathbb {R}^d\), there exists a solution of the focusing nonlinear Schrödinger equation with \(L^2\)-critical nonlinearity that blows up at the \(p\) points. The aim of this paper is to show that this result is still true if \(\mathbb {R}^d\) is replaced by a bounded and regular domain of \(\mathbb {R}^d\) with \(d=2,3\).

Reviewer: From the introduction.

MSC:

35Q55	NLS equations (nonlinear Schrödinger equations)
35B44	Blow-up in context of PDEs

Keywords:

nonlinear Schrödinger equation; blow up; \(L^2\)-critical nonlinearity

Citations:

Zbl 0707.35021