Electron. J. Diff. Equ., Vol. 2014 (2014), No. 74, pp. 1-11.

Pullback attractor for non-autonomous p-Laplacian equations with dynamic flux boundary conditions

Bo You, Fang Li

Abstract:
This article studies the long-time asymptotic behavior of solutions for the non-autonomous $p$-Laplacian equation
$$ u_t-\Delta_pu+ |u|^{p-2}u+f(u)=g(x,t) $$
with dynamic flux boundary conditions
$$ u_t+|\nabla u|^{p-2}\frac{\partial u}{\partial\nu}+f(u)=0 $$
in a n-dimensional bounded smooth domain $\Omega$ under some suitable assumptions. We prove the existence of a pullback attractor in $\big(W^{1,p}(\Omega)\cap L^q(\Omega)\big)\times L^q(\Gamma)$ by asymptotic a priori estimate.

Submitted June 26, 2013. Published March 18, 2014.
Math Subject Classifications: 35B40, 37B55.
Key Words: Pullback attractor; Sobolev compactness embedding; p-Laplacian; norm-to-weak continuous process; asymptotic a priori estimate; non-autonomous; nonlinear flux boundary conditions.

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Bo You
School of Mathematics and Statistics
Xi'an Jiaotong University
Xi'an, 710049, China
email: youb03@126.com
Fang Li
Department of Mathematics, Nanjing University
Nanjing, 210093, China
email: lifang101216@126.com

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