Global existence and blow-up for the focusing inhomogeneous nonlinear Schrödinger equation with inverse-square potential

Abstract

<p style='text-indent:20px;'>In this paper, we study the Cauchy problem for the focusing inhomogeneous nonlinear Schrödinger equation with inverse-square potential</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ iu_{t} +\Delta u-c|x|^{-2}u+|x|^{-b} |u|^{\sigma } u=0,\; u(0)=u_{0} \in H_{c}^{1},\;(t, x)\in \mathbb R\times\mathbb R^{d}, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ d\ge3 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ 0<b<2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ \frac{4-2b}{d}<\sigma<\frac{4-2b}{d-2} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ c>-c(d):=-\left(\frac{d-2}{2}\right)^{2} $\end{document}</tex-math></inline-formula>. We first establish the criteria for global existence and blow-up of general (not necessarily radial or finite variance) solutions to the equation. Using these criteria, we study the global existence and blow-up of solutions to the equation with general data lying below, at, and above the ground state threshold. Our results extend the global existence and blow-up results of Campos-Guzmán (Z. Angew. Math. Phys., 2021) and Dinh-Keraani (SIAM J. Math. Anal., 2021).</p>

Blow-up solutions of the intercritical inhomogeneous NLS equation: the non-radial case

“…Furthermore, they proved small data scattering, constructed wave operators in H 1 a and proved that suitable space-time bounds imply scattering in the intercritical case. Later on, An, Jan and Kim [3] extended the results of Duyckaerts and Roudenko [21] for the NLS and of the first and second authors [10] for the INLS to include a description of the global behavior of solutions to (INLS) a with mass-energy possibly greater than or equal to the ground state threshold.…”

“…In particular, it can be thought of as modeling inhomogeneities in the medium in which the wave propagates (see for instance [30]). The (INLS) a can thus be seen as a general model for various physical contexts, and it has been already studied in [11] and [3].…”

“…3 [M (t)] 2α 4 ≤ e 2α 4 λ v (τ i ), which implies that the annuli C i and C i+p(t) given by (4.28) are disjoints. In this case, for t close enough to T * , there are at leastN (t) 10p(t) disjoint annuli, thus (4.27) yields…”

Coupled inhomogeneous nonlinear Schrödinger system with potential in three space diemensions

“…The scalar nonlinear Schrödinger equation with inverse square potential, namely, (1.1) with m = 1 and u 𝑗 = 𝜆 |x| 2 u 𝑗 for 𝜆 > − 1 4 , was treated by previous studies [20][21][22], because it preserves the scaling invariance u 𝜅 ∶= 𝜅 2−𝜏 2(p−1) u (𝜅 2 •, 𝜅•), for 𝜅 > 0. This gives the classical dichotomy of global existence and scattering versus blow-up of solutions under the ground-state threshold in the inter-critical focusing regime [23].…”

“…The scalar nonlinear Schrödinger equation with inverse square potential, namely, () with $$$ m=1 $$$ and $$$ \mathcal{V}{u}_j=\frac{\lambda }{{\left|x\right|}^2}{u}_j $$$ for $$$ \lambda >-\frac{1}{4} $$$, was treated by previous studies [20–22], because it preserves the scaling invariance $$$ {u}_{\kappa}:= {\kappa}^{\frac{2-\tau }{2\left(p-1\right)}u}\left({\kappa}^2\cdotp, \kappa \cdotp \right) $$$, for $$$ \kappa >0 $$$. This gives the classical dichotomy of global existence and scattering versus blow‐up of solutions under the ground‐state threshold in the inter‐critical focusing regime [23].…”