论文标题
$ P <2 $的广义Hartree方程的加权空间中的适应性良好
Well-posedness in weighted spaces for the generalized Hartree equation with $p<2$
论文作者
论文摘要
我们研究了广义的hartree方程中的良好性$ iu_t +ΔU +（| x |^{ - （n-γ）} \ ast | u | u |^p）| u | u | u |^{p-2} u = 0 $，$ x \ in \ mathbb {r}我们根据Cazenave和Naumkin的思想建立了在加权Sobolev空间中的一类数据的局部良好性[6]。这至关重要的是，加权勒布斯格空间中Riesz转换的界限。结果，我们获得了一类全球存在的数据，此外，在积极的时间内分散了。此外，在$ l^2 $ - 整理环境中的聚焦案例中，我们以正能量获得了局部良好数据的子集，该数据在有限的时间内会爆炸。
We investigate the well-posedness in the generalized Hartree equation $iu_t + Δu + (|x|^{-(N-γ)} \ast |u|^p)|u|^{p-2}u=0$, $x \in \mathbb{R}^N$, $0<γ<N$, for low powers of nonlinearity, $p<2$. We establish the local well-posedness for a class of data in weighted Sobolev spaces, following ideas of Cazenave and Naumkin [6]. This crucially relies on the boundedness of the Riesz transform in weighted Lebesgue spaces. As a consequence, we obtain a class of data that exists globally, moreover, scatters in positive time. Furthermore, in the focusing case in the $L^2$-supercritical setting we obtain a subset of locally well-posed data with positive energy, which blows up in finite time.
