论文标题
低维度的一类不均匀的Biharmonic非线性schrödinger方程的能量散射
Energy scattering for a class of inhomogeneous biharmonic nonlinear Schrödinger equations in low dimensions
论文作者
论文摘要
我们考虑了一类Biharmonic非线性schrödinger方程，并具有焦点不均匀的功率型非线性\ [i \ partial_t u-Δ^2 u +μΔU +μΔU +| x | x |^{ - b} | u \ right | _ {t = 0} = u_0 \ in H^2（\ mathbb {r}^d）\] \ $ d \ geq 1，μ\ geq 0 $，$ 0 <b <\ sin \ {d，4 \}，4 \}，4 \} 5 $。我们首先确定一个区域，在该区域中，全球及时存在该方程的解决方案。然后，我们证明这些全局时间解决方案在三个及更高的维度中散布在$ h^2（\ mathbb {r}^d）$中。在没有谐波扰动的情况下，即$μ= 0 $，我们的结果扩展了Saanouni证明的能量散射[Calc。 var。 60（2021），艺术。不。 113]和坎波斯和古兹曼[Calc。 var。 61（2022），艺术。不。 156]至三个和四个维度。在存在排斥谐波扰动$μ> 0 $的情况下，我们的能量散射是新的。证明依赖于Lorentz空间中的估计值，这些估计值适合处理重量$ | x |^{ - b} $。
We consider a class of biharmonic nonlinear Schrödinger equations with a focusing inhomogeneous power-type nonlinearity \[ i\partial_t u -Δ^2 u+μΔu +|x|^{-b} |u|^αu=0, \quad \left. u\right|_{t=0}=u_0 \in H^2(\mathbb{R}^d) \] with $d\geq 1, μ\geq 0$, $0<b<\min\{d,4\}$, $α>0$, and $α<\frac{8-2b}{d-4}$ 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., $μ=0$, our result extends the energy scattering proved by Saanouni [Calc. Var. 60 (2021), art. no. 113] and Campos and Guzmán [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 $μ>0$. The proofs rely on estimates in Lorentz spaces which are properly suited for handling the weight $|x|^{-b}$.
