Pullback attractors for the non-autonomous quasi-linear complex Ginzburg-Landau equation with \(p\)-Laplacian.

*(English)*Zbl 1351.37261Summary: In this paper, we are concerned with the long-time behavior of the following non-autonomous quasi-linear complex Ginzburg-Landau equation with \(p\)-Laplacian
\[
\frac{\partial u}{\partial t}-(\lambda+i\alpha)\Delta_p u+(\kappa+i\beta)|u|^{q-2}u-\gamma u=g(x,t)
\]
without any restriction on \(q>2\) under additional assumptions. We first prove the existence of a pullback absorbing set in \(L^2(\Omega) \cap W^{1,p}_0(\Omega)\cap L^q(\Omega)\) for the process \(\{U(t,\tau)\}_{t\geq \tau}\) corresponding to the non-autonomous quasi-linear complex Ginzburg-Landau equation (1)–(3) with \(p\)-Laplacian. Next, the existence of a pullback attractor in \(L^2(\Omega)\) is established by the Sobolev compactness embedding theorem. Finally, we prove the existence of a pullback attractor in \(W^{1,p}_0(\Omega)\) for the process \(\{U(t,\tau)\}_{t\geq \tau}\) associated with the non-autonomous quasi-linear complex Ginzburg-Landau equation (1)–(3) with \(p\)-Laplacian by asymptotic a priori estimates.

##### MSC:

37L25 | Inertial manifolds and other invariant attracting sets of infinite-dimensional dissipative dynamical systems |

37L30 | Infinite-dimensional dissipative dynamical systems–attractors and their dimensions, Lyapunov exponents |

35B40 | Asymptotic behavior of solutions to PDEs |

35B41 | Attractors |

35B45 | A priori estimates in context of PDEs |

##### Keywords:

pullback attractor; Ginzburg-Landau equations; Sobolev compactness embedding theorem; asymptotic a priori estimates
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\textit{B. You} et al., Discrete Contin. Dyn. Syst., Ser. B 19, No. 6, 1801--1814 (2014; Zbl 1351.37261)

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