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+iα)Δu+(κ+iβ)|u|p−2u−γu=0 without any restriction on p>2p>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 L2(Ω)L2(Ω), H01(Ω)∩Lp(Ω) and H2(Ω)∩L2(p−1)(Ω)H2(Ω)∩L2(p−1)(Ω), respectively, for the semigroup {S(t)}t⩾0{S(t)}t⩾0 generated by (1.1)–(1.3) is established. Finally, we prove the existence of global attractors in L2(Ω)L2(Ω) and H01(Ω) for the semigroup {S(t)}t⩾0{S(t)}t⩾0 generated by (1.1)–(1.3) by the Sobolev compactness embedding theorem and prove the existence of global attractor in Lp(Ω)Lp(Ω) for the semigroup {S(t)}t⩾0{S(t)}t⩾0 generated by (1.1)–(1.3) using interpolation inequality.