Uniform attractors for non-autonomous nonclassical diffusion equations on $\mathbb R^N$
Bull. Korean Math. Soc. 2014 Vol. 51, No. 5, 1299-1324
https://doi.org/10.4134/BKMS.2014.51.5.1299
Printed September 30, 2014
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
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