Finite time blow-up and global existence for the nonlocal complex Ginzburg-Landau equation

In this paper, we consider the Cauchy problem for a family of nonlocal complex Ginzburg-Landau equatione−iθut=Δu+(1|x|n−α⁎|u|p)|u|p−2u−λu on Rn, 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 H2 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.