We consider the inhomogeneous nonlinear Schrödinger (INLS) equation in ${\mathbb{R}}^N$$i{\partial }_{t}u+\mathrm{\Delta }u+|x{|}^{-b}|u{|}^{2\sigma }u=0,$ where $N\ge 3$, $0<b<\min \{\frac{N}{2},2\}$ and $\frac{2-b}{N}<\sigma <\frac{2-b}{N-2}$. The scaling invariant Sobolev space is ${\dot{H}}^{s_c}$ with $s_c=\frac{N}{2}-\frac{2-b}{2\sigma }$. The restriction on σ implies $0<s_c<1$ and the equation is called intercritical (i.e. mass-supercritical and energy-subcritical). Let $u_0\in {\dot{H}}^{s_c}\cap {\dot{H}}^1$ be a radial initial data and $u(t)$ the corresponding solution to the INLS equation. We first show that if $E[u_0]\le 0$, then the maximal time of existence of the solution $u(t)$ is finite. Also, for all radially symmetric solution of the INLS equation with finite maximal time of existence $T^{⁎}>0$, then ${\lim \sup }_{t\to T^{⁎}}{\Vert u(t)\Vert }_{{\dot{H}}^{s_c}}=+\infty$. Moreover, under an additional assumption and recalling that ${\dot{H}}^{s_c}\subset L^{{\sigma }_c}$ with ${\sigma }_c=\frac{2N\sigma }{2-b}$, we can in fact deduce, for some $\gamma =\gamma (N,\sigma ,b)>0$, the following lower bound for the blow-up rate$c{\Vert u(t)\Vert }_{{\dot{H}}^{s_c}}\ge {\Vert u(t)\Vert }_{L^{{\sigma }_c}}\ge |\log (T-t){|}^{\gamma },\text{as}t\to T^{⁎}.$ The proof is based on the ideas introduced for the $L^2$ super critical nonlinear Schrödinger equation in the work of Merle and Raphaël [14] and here we extend their results to the INLS setting.

我们考虑非齐次非线性薛定谔 (INLS) 方程 ${\mathbb{电阻}}^N$$\mathrm{一世}{\partial }_{吨}你+\mathrm{\Delta }你+|X{|}^{-乙}|你{|}^{2\sigma }你=0,$ 在哪里 $N\ge 3$, $0<乙<\mathrm{分钟}\{\frac{N}{2},2\}$ 和 $\frac{2-乙}{N}<\sigma <\frac{2-乙}{N-2}$. 标度不变的 Sobolev 空间是${\dot{H}}^{{秒}_C}$ 和 ${秒}_C=\frac{N}{2}-\frac{2-乙}{2\sigma }$. 对σ的限制意味着$0<{秒}_C<1$该方程称为临界间（即质量-超临界和能量-亚临界）。让${你}_0\in {\dot{H}}^{{秒}_C}\cap {\dot{H}}^1$ 是径向初始数据和 $你(吨)$INLS 方程的相应解。我们首先证明如果$乙[{你}_0]\le 0$，那么解的最大存在时间 $你(吨)$是有限的。此外，对于具有有限最大存在时间的 INLS 方程的所有径向对称解${吨}^{⁎}>0$， 然后 ${\mathrm{林}\mathrm{超}}_{吨\to {吨}^{⁎}}{\Vert 你(吨)\Vert }_{{\dot{H}}^{{秒}_C}}=+\infty$. 此外，在一个额外的假设下，并回顾${\dot{H}}^{{秒}_C}\subset {升}^{{\sigma }_C}$ 和 ${\sigma }_C=\frac{2N\sigma }{2-乙}$，我们实际上可以推断，对于某些 $\gamma =\gamma (N,\sigma ,乙)>0$，以下爆破率的下限$C{\Vert 你(吨)\Vert }_{{\dot{H}}^{{秒}_C}}\ge {\Vert 你(吨)\Vert }_{{升}^{{\sigma }_C}}\ge |\mathrm{日志}(吨-吨){|}^{\gamma },\text{作为}吨\to {吨}^{⁎}.$ 证明是基于为 ${升}^2$ Merle 和 Raphaël [14] 的工作中的超临界非线性薛定谔方程，在这里我们将他们的结果扩展到 INLS 设置。