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A solution to the focusing 3d NLS that blows up on a contracting sphere. (English) Zbl 1325.35201

This paper concerns blow-up solutions of the cubic focusing nonlinear Schrödinger equations \[ i\partial_t\psi+\Delta\psi+2|\psi|^2\psi=0 \] in \(\mathbb R^3\). The authors construct a radial \(H^1\) solution \(\psi(x,t)\), which starts at \(t_0>0\), and evolves backwards in time \(t\), and blows up at \(t=0\). The blow-up date is \[ \|\nabla\psi(t)\|_{L^2(\mathbb R^3)}\sim t^{-2/3}. \] Moreover \[ \||x|\psi(t)\|_{L^2(\mathbb R^3)}\sim t^{1/3}, \] which indicates that the solution concentrates on a sphere of radius \(\sim t^{1/3}\). The solution is obtained by approximation, and the radial approximate solutions are constructed by using the ground state solution of the equation \[ \Delta\phi+\phi+2|\phi|^2\phi=0. \] It is well-known that, up to translations and up to a multiplier \(-1\), the ground state solution is unique, positive, radial and exponentially decaying. To get the true solution, good controls on the error terms are obtained by a very careful analysis.

MSC:

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

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