Blowup of $H^1$ solutions for a class of the focusing inhomogeneous nonlinear Schrödinger equation

Abstract

In this paper, we consider a class of the focusing inhomogeneous nonlinear Schrödinger equation \[ i\partial_t u + Δu + |x|^{-b} |u|^αu = 0, \quad u(0)=u_0 \in H^1(\mathbb{R}^d), \] 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 ${\alpha }^{\star }=\infty$ 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 $H^1\cap L^2(|x{|}^{2}dx)$ with $d\ge 1$.

19 pages, revised version, to appear in Nonlinear Analysis

References in corpus (2)

• Scattering theory in a weighted $L^2$ space for a class of the defocusing inhomogeneous nonlinear Schrödinger equation
• Scattering for the radial focusing INLS equation in higher dimensions

Cited by in corpus (2)

• A remark on the scattering theory for the 2d radial focusing INLS
• Scattering and blow-up criteria for 3D cubic focusing nonlinear inhomogeneous NLS with a potential