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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 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
35B40 Asymptotic behavior of solutions to PDEs
35B41 Attractors
35B45 A priori estimates in context of PDEs
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