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Blow-up in several points for the nonlinear Schrödinger equation on a bounded domain. (English) Zbl 1249.35303
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 \(d=2,3\).

MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
35B44 Blow-up in context of PDEs
Citations:
Zbl 0707.35021
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