Electron. J. Diff. Equ., Vol. 2016 (2016), No. 208, pp. 1-15.

Well-posedness of non-autonomous degenerate parabolic equations under singular perturbations

Jingyu Wang, Yejuan Wang, Dun Zhao

Abstract:
This article concerns the asymptotic behavior of the following non-autonomous degenerate parabolic equation with singular perturbations defined on a bounded domain in $\mathbb{R}^n$,
$$ \frac{\partial u}{\partial t}+\lambda u-\hbox{div}(|\nabla u|^{p-2}\nabla u) -\varepsilon \hbox{div}\Big(\big|\nabla \frac{\partial u}{\partial t}\big|^{p-2} \nabla \frac{\partial u}{\partial t}\Big) +f(x,t,u)=g(x,t), $$
where $\lambda$ is a positive constant, $p>2$ and $\varepsilon\in(0,1]$. The well-posedness and upper semicontinuity of pullback attractors are established for the problem without the uniqueness of solutions under singular perturbations.

Submitted August 3, 2015. Published August 2, 2016.
Math Subject Classifications: 34K26, 35B41, 35K65.
Key Words: Pullback attractor; multi-valued process; upper semicontinuity; p-laplacian equation; singular perturbation.

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  Jingyu Wang
School of Mathematics and Statistics
Gansu Key Laboratory of Applied Mathematics and Complex Systems
Lanzhou University, Lanzhou 730000, China
email: wangjy418@qq.com
Yejuan Wang
School of Mathematics and Statistics
Gansu Key Laboratory of Applied Mathematics and Complex Systems
Lanzhou University, Lanzhou 730000, China
email: wangyj@lzu.edu.cn
  Dun Zhao
School of Mathematics and Statistics
Gansu Key Laboratory of Applied Mathematics and Complex Systems
Lanzhou University, Lanzhou 730000, China
email: zhaod@lzu.edu.cn

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