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We present weighted estimates and higher order asymptotic expansions ofglobal solutions to the complex Ginzburg--Landau (CGL) type equation in thesuper Fujita-critical case. Our approach is based on commutation relationsbetween the CGL semigroup and monomial weights in $\mathbb{R}^{n}$ for theweighted estimates and on the Taylor expansions with respect to the both spaceand time variables for the asymptotic expansions. We also characterize theoptimal decay rate in time of the remainder for the asymptotic expansion fromthe viewpoint of the moments of the initial data in space and those of thenonlinear term in spacetime.

Asymptotic behavior of global solutions to the complex Ginzburg--Landau type equation in the super Fujita-critical case