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(0)(-i omega)(C) D(t)(alpha)u - Delta u = e(i gamma)vertical bar u vertical bar(p-1)u, x is an element of R-N, t > 0, u(0,x) = u(0)(x), x is an element of R-N, where 0 < alpha < 1, gamma is an element of R, -pi + pi alpha/2 < w < pi - pi alpha/2 < omega < pi - pi alpha/2, p > 1, u(0) is an element of L-q(R-N) (q >= q(c) = N(P-1)/2 and q >= 1) is a complex-valued function, and (C)(0)D(t)(alpha)u = partial derivative/partial derivative t(0)I(t)(1-alpha) (u(t,x) - u(0, x)), where I-0(t)1-alpha denotes a left Riemann-Liouville fractional integral of order 1 - alpha. 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], L-q (R-N)) and L2 gamma q/alpha N(gamma-q)((0, T), L-gamma (R-N)), where gamma satisfies 1/q - 1/r < 2/N. Moreover, we also obtain the existence of global solutions when parallel to u(0)parallel to Lqc(R-N) is sufficiently small. (C) 2018 Elsevier Inc. All rights reserved.