Dynamics for a class of non-autonomous degenerate \(p\)-Laplacian equations. (English) Zbl 1476.35123
Summary: In this paper, we investigate a class of non-autonomous degenerate \(p\)-Laplacian equations
\[
\partial_tu-\operatorname{div}(a(x)|\nabla u|^{p-2}\nabla u)+\lambda u+f(u)=g(x,t)
\]
in \(\Omega\), where \(a(x)\) is allowed to vanish on a nonempty closed subset with Lebesgue measure zero, \(g(x,t)\in L_{\text{loc}}^{p'}(\mathbb{R};D^{-1,p'}(\Omega,a))\) and \(\Omega\) an unbounded domain in \(\mathbb{R}^N\). We first establish the well-posedness of these equations by constructing a compact embedding. Then we show the existence of the minimal pullback \(\mathcal{D}_\mu\)-attractor, and prove that it indeed attracts the \(\mathcal{D}_\mu\) class in \(L^{2+\delta}\)-norm for any \(\delta\in[0,\infty)\). Our results extend some known ones in previously published papers.
MSC:
| 35K59 | Quasilinear parabolic equations |
| 35K92 | Quasilinear parabolic equations with \(p\)-Laplacian |
| 35B41 | Attractors |
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
| 35K65 | Degenerate parabolic equations |
Keywords:
non-autonomous; degenerate \(p\)-Laplacian equations; pullback attractors; higher-order attractionReferences:
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