论文标题
对径向溶液的爆炸，用于中间临界NLS方程
Blow-up of radial 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， $ 0 <b <\ sin \ left \ {\ frac {n} {2}，2 \ right \} $和$ \ frac {2-b} {n} {n} <σ<\ frac {2-b} {2-b} {n-2} $。缩放不变的sobolev space是$ \ dot {h}^{s_c} $，带有$ s_c = \ frac {n} {2} {2} - \ frac {2-b} {2σ} $。对$σ$的限制意味着$ 0 <s_c <1 $，方程称为中批评（即质量 - 占地和能量占领）。令$ u_0 \ in \ dot h^{s_c} \ cap \ dot h^1 $为径向初始数据，$ u（t）$是INLS方程的相应解决方案。我们首先表明，如果$ e [u_0] \ leq 0 $，那么解决方案$ u（t）$的最大时间是有限的。另外，对于INLS方程的所有径向对称解，具有有限的生存时间$ t^{\ ast}> 0 $，然后$ \ limSup_ {此外，在一个额外的假设下，回忆起$ \ dot {h}^{s_c} \ subset l^{σ_c} $带有$σ_c= \ frac {2nσ} {2nσ} {2-b} $，实际上我们可以推迟一些$γ=γ=γ=γ（n，f Blow），以下是blow的量$ c c \ | u（t）\ | _ {\ dot h^{s_c}} \ geq \ | u（t） t^{\ ast}。$$证明是基于$ l^2 $超关键的非线性schrödinger方程所介绍的想法，在Merle和Raphaël[13]的工作中，我们将其结果扩展到INLS设置。
We consider the inhomogeneous nonlinear Schrödinger (INLS) equation in $\mathbb{R}^N$ $$i \partial_t u +Δu +|x|^{-b} |u|^{2σ}u = 0,$$ where $N\geq 3$, $0<b<\min\left\{\frac{N}{2},2\right\}$ and $\frac{2-b}{N}<σ<\frac{2-b}{N-2}$. The scaling invariant Sobolev space is $\dot{H}^{s_c}$ with $s_c=\frac{N}{2}-\frac{2-b}{2σ}$. The restriction on $σ$ implies $0<s_c<1$ and the equation is called intercritical (i.e. mass-supercritical and energy-subcritical). Let $u_0\in \dot H^{s_c}\cap \dot H^1$ be a radial initial data and $u(t)$ the corresponding solution to the INLS equation. We first show that if $E[u_0]\leq 0$, then the maximal time of existence of the solution $u(t)$ is finite. Also, for all radially symmetric solution of the INLS equation with finite maximal time of existence $T^{\ast}>0$, then $\limsup_{t\rightarrow T^{\ast}}\|u(t)\|_{\dot H^{s_c}}=+\infty$. Moreover, under an additional assumption and recalling that $\dot{H}^{s_c} \subset L^{σ_c}$ with $σ_c=\frac{2Nσ}{2-b}$, we can in fact deduce, for some $γ=γ(N,σ,b)>0$, the following lower bound for the blow-up rate $$c\|u(t)\|_{\dot H^{s_c}}\geq \|u(t)\|_{L^{σ_c}}\geq |\log (T-t)|^γ,\,\,\,\mbox{ as }\,\,\,t\rightarrow T^{\ast}.$$ The proof is based on the ideas introduced for the $L^2$ super critical nonlinear Schrödinger equation in the work of Merle and Raphaël [13] and here we extend their results to the INLS setting.
