Finite time/Infinite time blow-up behaviors for the inhomogeneous nonlinear Schr\"{o}dinger equation

Abstract

In this work, we consider the following focusing inhomogeneous nonlinear Schr\"odinger equation \begin{align*} i\partial_t u+\Delta u +|x|^{-b}|u|^p u=0,\quad (t, x)\in\mathbb{R}\times\mathbb{R}^N \end{align*} with $0<b<\mbox{min}\{2, N\}$ and $\frac{4-2b}{N}<p<\frac{4-2b}{N-2}$. Assume that $u_0 \in H^{1}(\mathbb{R}^N)$ and beyond the ground state threshold, then we prove the following two statements, (1) when $\frac{4-2b}{N}<p< \min\{\frac{4}{N}, \frac{4-2b}{N-2}\}$, or $p =\frac{4}{N}$ when $b \in (0, \frac 4 N)$, then the corresponding solution blows up in finite time; (2) when $\frac{4}{N}<p<\frac{4-2b}{N-2}$, we prove the finite or infinite time blow-up. Moreover, we can further obtain a precise lower bound of infinite time blow-up rate, that is \begin{equation*} \sup_{t\in[0,T]}\|\nabla u(t)\|_{L^2}\gtrsim T^{\kappa},\quad \mbox{for some} \quad \kappa>0. \end{equation*} To our knowledge, the statement (1) establishes the first finite time blow-up result for this equation in the intercritical case when the initial data $u_0$ doesn't have finite variance and is non-radial. The statement (2) gives the first result for the infinite time blow-up rate for this equation.