论文标题
关于中间批判性不均匀NLS方程的爆炸解决方案的稳定性和浓度
On well-posedness and concentration of blow-up solutions for the intercritical inhomogeneous NLS equation
论文作者
论文摘要
我们考虑$ \ Mathbb {r}^n $ $ $ $ $ i \ partial_t u + u +ΔU + | x | x | x |^{ - b} | u | U |^{2σ} u = 0，$ n \ geq 2 $ n \ geq 2 $和$ b> 0。我们首先获得了$ h^1 $的小数据全球结果，在两个空间维度中，该结果在$ b $范围内改善了[22]的第三作者结果。对于$ n \ geq 3 $和$ \ frac {2-b} {n} <σ<\ frac {2-b} {n-2} $，我们还研究了$ \ dot h^{s_c} \ cap \ cap \ dot h^1 $的本地良好姿势$ s_c = \ frac {n} {2} - \ frac {2-b} {2σ} $。使用gagliardo-nirenberg类型的估计，还建立了$ \ dot h^{s_c} \ cap \ dot h^1 $中全球解决方案的足够条件。最后，我们研究$ l^{σ_c} - $ norm浓度现象，其中$σ_c= \ frac {2nσ} {2-b} $，用于有限的时间爆破解决方案，$ \ dot h^{s_c} \ cap} \ cap \ cap \ dot h^1 $ and bound b bounded $ \ dot $ \ dot $ \ dot h^s_c} s_c_ = s_c} - $我们的方法基于$ \ dot h^{s_c} \ cap \ dot h^1 $的紧凑嵌入到加权$ l^{2σ+2} $ space中。
We consider the focusing inhomogeneous nonlinear Schrödinger (INLS) equation in $\mathbb{R}^N$ $$i \partial_t u +Δu + |x|^{-b} |u|^{2σ}u = 0,$$ where $N\geq 2$ and $σ$, $b>0$. We first obtain a small data global result in $H^1$, which, in the two spatial dimensional case, improves the third author result in [22] on the range of $b$. For $N\geq 3$ and $\frac{2-b}{N}<σ<\frac{2-b}{N-2}$, we also study the local well posedness in $\dot H^{s_c}\cap \dot H^1 $, where $s_c=\frac{N}{2}-\frac{2-b}{2σ}$. Sufficient conditions for global existence of solutions in $\dot H^{s_c}\cap \dot H^1$ are also established, using a Gagliardo-Nirenberg type estimate. Finally, we study the $L^{σ_c}-$norm concentration phenomenon, where $σ_c=\frac{2Nσ}{2-b}$, for finite time blow-up solutions in $\dot H^{s_c}\cap \dot H^1$ with bounded $\dot H^{s_c}-$norm. Our approach is based on the compact embedding of $\dot H^{s_c}\cap \dot H^1$ into a weighted $L^{2σ+2}$ space.
