On well-posedness and concentration of blow-up solutions for the intercritical inhomogeneous NLS equation

Abstract

We consider the focusing inhomogeneous nonlinear Schrödinger (INLS) equation in ${\mathbb{R}}^N$ $$i{\partial }_{t}u+\Delta u+|x{|}^{-b}|u{|}^{2\sigma }u=0,$$ where $N\ge 2$ and $\sigma$, $b>0$. We first obtain a small data global result in $H^1$, which, in the two spatial dimensional case, improves the third author result in [22] on the range of $b$. For $N\ge 3$ and $\frac{2-b}{N}<\sigma <\frac{2-b}{N-2}$, we also study the local well posedness in ${\dot{H}}^{s_c}\cap {\dot{H}}^1$, where $s_c=\frac{N}{2}-\frac{2-b}{2\sigma }$. Sufficient conditions for global existence of solutions in ${\dot{H}}^{s_c}\cap {\dot{H}}^1$ are also established, using a Gagliardo-Nirenberg type estimate. Finally, we study the $L^{{\sigma }_c}-$norm concentration phenomenon, where ${\sigma }_c=\frac{2N\sigma }{2-b}$, for finite time blow-up solutions in ${\dot{H}}^{s_c}\cap {\dot{H}}^1$ with bounded ${\dot{H}}^{s_c}-$norm. Our approach is based on the compact embedding of ${\dot{H}}^{s_c}\cap {\dot{H}}^1$ into a weighted $L^{2\sigma +2}$ space.

arXiv comment27 pages

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Cited by in corpus (2)

• Blow-up of radial solutions for the intercritical inhomogeneous NLS equation
• On the inhomogeneous NLS with inverse-square potential