Local and global existence in L^{p} for the inhomogeneous nonlinear Schrödinger equation

This paper investigates the local and global existence for the inhomogeneous nonlinear Schrödinger equation with the nonlinearity λ|x|^{-b}|u|^{β}u. It is show that a global solution exists in the mass-subcritical for large data in the spaces L^{p}, p < 2 under some suitable conditions on b,β and p. The solution is established using a data-decomposition argument, two kinds of generalized Strichartz estimates in Lorentz spaces and a interpolation theorem.

Energy scattering for a class of inhomogeneous nonlinear Schrödinger equation in two dimensions

We consider a class of [Formula: see text]-supercritical inhomogeneous nonlinear Schrödinger equations in two dimensions [Formula: see text] where [Formula: see text] and [Formula: see text]. Using a new approach of Arora et al. [Scattering below the ground state for the 2D radial nonlinear Schrödinger equation, Proc. Amer. Math. Soc. 148 (2020) 1653–1663], we show the energy scattering for the equation with radially symmetric initial data. In the focusing case, our result extends the one of Farah and Guzmán [Scattering for the radial focusing INLS equation in higher dimensions, Bull. Braz. Math. Soc. (N.S.) 51 (2020) 449–512] to the whole range of [Formula: see text] where the local well-posedness is available. In the defocusing case, our result extends the one in [V. D. Dinh, Energy scattering for a class of the defocusing inhomogeneous nonlinear Schrödinger equation, J. Evol. Equ. 19(2) (2019) 411–434], where the energy scattering for non-radial initial data was established in dimensions [Formula: see text].

Sharp condition of global well-posedness for inhomogeneous nonlinear Schrödinger equation

<p style='text-indent:20px;'>This paper studies the Cauchy problem of Schrödinger equation with inhomogeneous nonlinear term <inline-formula><tex-math id="M1">\begin{document}$ V(x)|\varphi|^{p-1}\varphi $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula>. For the case <inline-formula><tex-math id="M3">\begin{document}$ p&gt;1+\frac{4(1+\varepsilon_0)}{n} (0&lt;\varepsilon_0&lt;\frac{2}{n-2}) $\end{document}</tex-math></inline-formula>, by introducing a potential well, we obtain some invariant sets of solution and give a sharp condition of global existence and finite time blowup of solution; for the case <inline-formula><tex-math id="M4">\begin{document}$ p&lt;1+\frac{4}{n} $\end{document}</tex-math></inline-formula>, we obtain the global existence of solution for any initial data in <inline-formula><tex-math id="M5">\begin{document}$ H^1 (\mathbb{R}^n) $\end{document}</tex-math></inline-formula>.</p>