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Pullback attractors for the non-autonomous quasi-linear complex Ginzburg-Landau equation with $p$-Laplacian

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  • 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 \begin{align*} \frac{\partial u}{\partial t}-(\lambda+i\alpha)\Delta_p u+(\kappa+i\beta)|u|^{q-2}u-\gamma u=g(x,t) \end{align*} 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.
    Mathematics Subject Classification: Primary: 37B55; Secondary: 37N20.

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