Abstract: We first used Sobolev inequalities and energy estimates to prove the global existence of solutions to the four-dimensional cubic complex Ginzburg-Landau equation. Then, we used the Gronwall lemma and Holder’s inequality to prove the existence of the absorption set of the equation and gave both the existence and compact connectedness of the maximal attractor of the equation.

Key words: complex Ginzburg-Landau equation, attractor, Sobolev inequality, Gronwall lemma