Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2014; 51(5): 1299-1324

Printed September 30, 2014

https://doi.org/10.4134/BKMS.2014.51.5.1299

Copyright © The Korean Mathematical Society.

Uniform attractors for non-autonomous nonclassical diffusion equations on $\mathbb R^N$

Cung The Anh and Nguyen Duong Toan

Hanoi National University of Education, Haiphong University

Abstract

We prove the existence of uniform attractors $\mathcal A_{\varepsilon}$ in the space $H^1(\mathbb{R}^N)\cap L^p(\mathbb{R}^N)$ for the following non-autonomous nonclassical diffusion equations on $\mathbb R^N$,\begin{equation*} u_t - \varepsilon \Delta u_t - \Delta u + f(x,u)+\lambda u = g(x,t),~ \varepsilon\in (0,1]. \end{equation*} The upper semicontinuity of the uniform attractors $\{\mathcal A_{\varepsilon}\}_{\varepsilon\in [0,1]}$ at $\varepsilon = 0$ is also studied.

Keywords: nonclassical diffusion equation, uniform attractor, unbounded domain, upper semicontinuity, tail estimates method, asymptotic a priori estimate method

MSC numbers: 35B41, 35K70, 35D30