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Pullback attractors for the non-autonomous quasi-linear complex Ginzburg-Landau equation with $$p$$-Laplacian. (English) Zbl 1351.37261
Summary: 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
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