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Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation

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  • Using variational methods we study the stability and strong instability of ground states for the focusing inhomogeneous nonlinear Schrödinger equation (INLS)

    $ \begin{equation*} i\partial_{t}u+\Delta u+|x|^{-b}|u|^{p-1}u = 0. \end{equation*} $

    We construct two kinds of invariant sets under the evolution flow of (INLS). Then we show that the solution of (INLS) is global and bounded in $ H^{1}(\mathbb{R^{N}}) $ in the first kind of the invariant sets, while the solution blow-up in finite time in the other invariant set. Consequently, we prove that if the nonlinearity is $ L^{2} $-supercritical, then the ground states are strongly unstable by blow-up.

    Mathematics Subject Classification: Primary: 35Q55; Secondary: 35B44.


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