Blowup of $H^1$ solutions for a class of the focusing inhomogeneous nonlinear Schr\"odinger equation

Abstract

In this paper, we consider a class of the focusing inhomogeneous nonlinear Schr\"odinger equation \[ i\partial_t u + \Delta u + |x|^{-b} |u|^\alpha u = 0, \quad u(0)=u_0 \in H^1(\mathbb{R}^d), \] with $0<b<\min\{2,d\}$ and $\alpha_\star\leq \alpha <\alpha^\star$ where $\alpha_\star =\frac{4-2b}{d}$ and $\alpha^\star=\frac{4-2b}{d-2}$ if $d\geq 3$ and $\alpha^\star = \infty$ if $d=1,2$. In the mass-critical case $\alpha=\alpha_\star$, we prove that if $u_0$ has negative energy and satisfies either $xu_0 \in L^2$ with $d\geq 1$ or $u_0$ is radial with $d\geq 2$, then the corresponding solution blows up in finite time. Moreover, when $d=1$, we prove that if the initial data (not necessarily radial) has negative energy, then the corresponding solution blows up in finite time. In the mass and energy intercritical case $\alpha_\star< \alpha <\alpha^\star$, we prove the blowup below ground state for radial initial data with $d\geq 2$. This result extends the one of Farah in \cite{Farah} where the author proved blowup below ground state for data in the virial space $H^1\cap L^2(|x|^2 dx)$ with $d\geq 1$.