Global attractors for the complex Ginzburg–Landau equation

This paper concerns the long-time behavior of the following complex Ginzburg–Landau equations ∂ u ∂ t − ( λ + i α ) Δ u + ( κ + i β ) | u | p − 2 u − γ u = 0 without any restriction on p > 2 under the assumptions (1.4). We first prove the well-posedness of strong solutions for the complex Ginzburg–Landau equations, and then the existence of absorbing sets in L 2 ( Ω ) , H 0 1 ( Ω ) ∩ L p ( Ω ) and H 2 ( Ω ) ∩ L 2 ( p − 1 ) ( Ω ) , respectively, for the semigroup { S ( t ) } t ⩾ 0 generated by (1.1)–(1.3) is established. Finally, we prove the existence of global attractors in L 2 ( Ω ) and H 0 1 ( Ω ) for the semigroup { S ( t ) } t ⩾ 0 generated by (1.1)–(1.3) by the Sobolev compactness embedding theorem and prove the existence of global attractor in L p ( Ω ) for the semigroup { S ( t ) } t ⩾ 0 generated by (1.1)–(1.3) using interpolation inequality.