Blow-up criteria for the 3d cubic nonlinear Schrödinger equation

Abstract

We consider solutions $u$ to the 3d nonlinear Schrödinger equation $i{\partial }_{t}u+\Delta u+|u{|}^{2}u=0$. In particular, we are interested in finding criteria on the initial data $u_0$ that predict the asymptotic behavior of $u(t)$, e.g., whether $u(t)$ blows-up in finite time, exists globally in time but behaves like a linear solution for large times (scatters), or exists globally in time but does not scatter. This question has been resolved (at least for $H^1$ data) if $M[u]E[u]\le M[Q]E[Q]$, where $M[u]$ and $E[u]$ denote the mass and energy of $u$, and $Q$ denotes the ground state solution to $-Q+\Delta Q+|Q{|}^{2}Q=0$. Here, we prove a new sufficient condition for blow-up using an interpolation type inequality and the virial identity that is applicable to certain initial data satisfying $M[u]E[u]>M[Q]E[Q]$. Our condition is similar to one obtained by Lushnikov (1995) but our method allows for an adaptation to radial, infinite-variance initial data that can be stated conceptually: for real initial data, if a certain fraction of the mass is contained in the unit ball, then blow-up occurs. We also show analytically (if one takes the numerically computed value of $\Vert Q{\Vert }_{{\dot{H}}^{1/2}}$) that there exist Gaussian initial data $u_0$ with negative quadratic phase such that $\Vert u_0{\Vert }_{{\dot{H}}^{1/2}}<\Vert Q{\Vert }_{{\dot{H}}^{1/2}}$ but the solution $u(t)$ blows-up. We conclude with several numerically computed examples.

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Cited by in corpus (1)

• Standing Ring Blowup Solutions for Cubic NLS