Blow-up of radial solutions for the intercritical inhomogeneous NLS equation

Abstract

We consider the inhomogeneous nonlinear Schr\"odinger (INLS) equation in $\mathbb{R}^N$ $$i \partial_t u +\Delta u +|x|^{-b} |u|^{2\sigma}u = 0,$$ where $N\geq 3$, $0<b<\min\left\{\frac{N}{2},2\right\}$ 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 $\sigma$ 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 $E[u_0]\leq 0$, 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 $T^{\ast}>0$, then $\limsup_{t\rightarrow T^{\ast}}\|u(t)\|_{\dot H^{s_c}}=+\infty$. 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 $$c\|u(t)\|_{\dot H^{s_c}}\geq \|u(t)\|_{L^{\sigma_c}}\geq |\log (T-t)|^{\gamma},\,\,\,\mbox{ as }\,\,\,t\rightarrow T^{\ast}.$$ The proof is based on the ideas introduced for the $L^2$ super critical nonlinear Schr\"odinger equation in the work of Merle and Rapha\"el [13] and here we extend their results to the INLS setting.