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

In this paper, we consider a class of the focusing inhomogeneous nonlinear Schr"odinger equation $$i{\mathrm{\partial }}_{t}u+\mathrm{\Delta }u+\mathrm{\mid }x{\mathrm{\mid }}^{-b}\mathrm{\mid }u{\mathrm{\mid }}^{\alpha }u=0,u(0)=u_0\in H^1({\mathbb{R}}^d),$$ with $0<b<\min \{2,d\}$ and ${\alpha }_{\star }\le \alpha <{\alpha }^{\star }$ where ${\alpha }_{\star }=\frac{4-2b}{d}$ and ${\alpha }^{\star }=\frac{4-2b}{d-2}$ if $d\ge 3$ and ${\alpha }^{\star }=\mathrm{\infty }$ if $d=1,2$. In the mass-critical case $\alpha ={\alpha }_{\star }$, we prove that if $u_0$ has negative energy and satisfies either $x{u}_0\in L^2$ with $d\ge 1$ or $u_0$ is radial with $d\ge 2$, 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 $d\ge 2$. This result extends the one of Farah in \cite{Farah} where the author proved blowup below ground state for data in the virial space $H^1\cap L^2(\mathrm{\mid }x{\mathrm{\mid }}^{2}dx)$ with $d\ge 1$.