Finite time blow-up and global existence for the nonlocal complex Ginzburg-Landau equation
Finite time blow-up and global existence for the nonlocal complex Ginzburg-Landau equation
复制标题
非局部复数Ginzburg-Landau方程的有限时间爆炸和全局存在性
DOI：
10.1016/j.jmaa.2018.06.038
复制
发表时间：
2018
影响因子：
1.3
通讯作者：
Liu Baiyu
中科院分区：
文献类型：
--
作者：
Li Xiaoliang;Liu Baiyu
本文考虑一族非局部复Ginzburg-Landau方程e−iθut=Δu+(1|x|n−α⁎|u|p)|u|p−2 u−λu在Rn上的柯西问题，其中−π/2<；θ<；π/2和λ∈R.然后建立了H2正则性，导出了能量恒等式。通过构造一些不变集，进一步得到了解的有限时间爆破和整体存在的一些充分条件，特别是确定了当λ>；0时初值的一个尖锐阈值。此外，我们估计了解的寿命作为θ的函数，并得到了λ∈R的最大生存时间的下界。
In this paper, we consider the Cauchy problem for a family of nonlocal complex Ginzburg–Landau equation e− i θ u t= Δ u+(1| x| n− α⁎| u| p)| u| p− 2 u− λ u on R n, where− π/2< θ< π/2 and λ∈ R. First, the local well-posedness in Lebesgue spaces is established by a fixed point argument. Then we set up the H 2 regularity and derive the energy identities. By constructing some invariant sets, we further find some sufficient conditions on finite time blow-up and global existence for solutions which in particular determines a sharp threshold of initial date when λ> 0. In addition, we estimate the lifespan of solutions as a function of θ and obtain the lower bound of the maximal existence time for λ∈ R.