Blowup of H1 solutions for a class of the focusing inhomogeneous nonlinear Schrodinger equation
Résumé
In this paper, we consider a class of the focusing inhomogeneous nonlinear Schrödinger equation $$i\partial_tu + \Delta u + |x|^{-b} ||u|^\alpha u = 0, u(0) = u_0 ∈ H^1 (\mathbb{R}^d),$$ 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 $\alpha = \infty$ 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 [9] where the author proved blowup below ground state for data in the virial space $H^1 \cap L^2 (|x|^2 dx)$ with $d ≥ 1$.
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