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Blow-up of non-radial solutions for the \(L^2\) critical inhomogeneous NLS equation. (English) Zbl 1492.35299

Summary: We consider the \(L^2\) critical inhomogeneous nonlinear Schrödinger equation in \(\mathbb{R}^N\,i\partial_tu+\Delta u+|x|^{-b}|u|^{\frac{4-2b}{N}}u=0\) where \(N\geq 1\) and \(0<b<\min\{2,N\}\). We prove that if \(u_0\in H^1(\mathbb{R}^N)\) satisfies \(E[u_0]<0\), then the corresponding solution blows-up in finite time. This is in sharp contrast to the classical \(L^2\) critical nonlinear Schrödinger equation where this type of result is only known in the radial case for \(N\geq 2\).

MSC:

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

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