论文标题
全球良好的和不均匀的Biharmonic NLS的临界标准浓度
Global well-posedness and critical norm concentration for inhomogeneous biharmonic NLS
论文作者
论文摘要
我们考虑$ \ Mathbb {r}^n $，$$ i \ partial_t u +u +δ^2 u- | x | x | x |^{ - b} |^e |^u | |^{2σ} u = 0，$ = 0，$ = 0 $ n $ c $ n $ n = 0 $ b> 0 $ b> 0。我们首先研究了$ \ dot h^{s_c} \ cap \ dot h^2 $，对于$ n \ geq 5 $和$ 0 <s_c <2 $，其中$ s_c = \ frac {n} {2} {2} {2} - \ frac {4-b} $。接下来，我们建立了一个gagliardo-nirenberg类型的不等式，以便获得足够的条件，以$ \ dot h^{s_c} \ cap \ dot h^2 $，带有$ 0 \ leq s_c s_c <2 $。最后，我们研究了$ l^{σ_c} $的现象 - 有限时间的规范浓度爆炸了解决方案，使用有限的$ \ dot h^{s_c} $ - norm，其中$σ_C= \ frac {2nσ}} {4-b} $。我们的主要工具是$ \ dot l^p \ cap \ dot h^2 $的紧凑嵌入到加权$ l^{2σ+2} $ space中，可以看到独立利益。
We consider the inhomogeneous biharmonic nonlinear Schrödinger (IBNLS) equation in $\mathbb{R}^N$, $$i \partial_t u +Δ^2 u -|x|^{-b} |u|^{2σ}u = 0,$$ where $σ>0$ and $b>0$. We first study the local well-posedness in $\dot H^{s_c}\cap \dot H^2 $, for $N\geq 5$ and $0<s_c<2$, where $s_c=\frac{N}{2}-\frac{4-b}{2σ}$. Next, we established a Gagliardo-Nirenberg type inequality in order to obtain sufficient conditions for global existence of solutions in $\dot H^{s_c}\cap \dot H^2$ with $0\leq s_c<2$. Finally, we study the phenomenon of $L^{σ_c}$-norm concentration for finite time blow up solutions with bounded $\dot H^{s_c}$-norm, where $σ_c=\frac{2Nσ}{4-b}$. Our main tool is the compact embedding of $\dot L^p\cap \dot H^2$ into a weighted $L^{2σ+2}$ space, which may be seen of independent interest.
