The local and global existence of solutions for a time fractional complex Ginzburg-Landau equation

We study the following time fractional complex nonlinear Ginzburg-Landau equation:{e−iω0CDtαu−△u=eiγ|u|p−1u,x∈RN,t>0,u(0,x)=u0(x),x∈RN, where 0<α<1, γ∈R, −π+πα2<ω<π−πα2, p>1, u0∈Lq(RN) (q≥qc=N(p−1)2 and q≥1) is a complex-valued function, and Dtα0Cu=∂∂t0It1−α(u(t,x)−u(0,x)), where It1−α0 denotes a left Riemann-Liouville fractional integral of order 1−α. By defining two operators and establishing some estimates of them, we prove the well-posedness of the mild solution for this problem in C([0,T],Lq(RN)) and L2rqαN(r−q)((0,T),Lr(RN)), where r satisfies 1/q−1/r<2/N. Moreover, we also obtain the existence of global solutions when ‖u0‖Lqc(RN) is sufficiently small.