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 \({\mathbb {R}}^N\)$$\begin{aligned} i \partial _t u +\Delta u + |x|^{-b} |u|^{2\sigma }u = 0, \end{aligned}$$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.