Energy scattering for a class of inhomogeneous biharmonic nonlinear Schr\"odinger equations in low dimensions

Abstract

We consider a class of biharmonic nonlinear Schr\"odinger equations with a focusing inhomogeneous power-type nonlinearity \[ i\partial_t u -\Delta^2 u+\mu\Delta u +|x|^{-b} |u|^\alpha u=0, \quad \left. u\right|_{t=0}=u_0 \in H^2(\mathbb{R}^d) \] with $d\geq 1, \mu\geq 0$, $0<b<\min\{d,4\}$, $\alpha>0$, and $\alpha<\frac{8-2b}{d-4}$ if $d\geq 5$. We first determine a region in which solutions to the equation exist globally in time. We then show that these global-in-time solutions scatter in $H^2(\mathbb{R}^d)$ in three and higher dimensions. In the case of no harmonic perturbation, i.e., $\mu=0$, our result extends the energy scattering proved by Saanouni [Calc. Var. 60 (2021), art. no. 113] and Campos and Guzm\'an [Calc. Var. 61 (2022), art. no. 156] to three and four dimensions. Our energy scattering is new in the presence of a repulsive harmonic perturbation $\mu>0$. The proofs rely on estimates in Lorentz spaces which are properly suited for handling the weight $|x|^{-b}$.