## 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

Zbl 0707.35021
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