| Popis: |
We study the following time fractional complex nonlinear Ginzburg–Landau equation: { e − i ω 0 C D t α u − △ u = e i γ | u | p − 1 u , x ∈ R N , t > 0 , u ( 0 , x ) = u 0 ( x ) , x ∈ R N , where 0 α 1 , γ ∈ R , − π + π α 2 ω π − π α 2 , p > 1 , u 0 ∈ L q ( R N ) ( q ≥ q c = N ( p − 1 ) 2 and q ≥ 1 ) is a complex-valued function, and D t α 0 C u = ∂ ∂ t 0 I t 1 − α ( u ( t , x ) − u ( 0 , x ) ) , where I t 1 − α 0 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 ] , L q ( R N ) ) and L 2 r q α N ( r − q ) ( ( 0 , T ) , L r ( R N ) ) , where r satisfies 1 / q − 1 / r 2 / N . Moreover, we also obtain the existence of global solutions when ‖ u 0 ‖ L q c ( R N ) is sufficiently small. |
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