Dynamics of non-autonomous nonclassical diffusion equations on $R^n$

Cung The Anh; Tang Quoc Bao

doi:10.3934/cpaa.2012.11.1231

Article Contents

2012, Volume 11, Issue 3: 1231-1252. Doi: 10.3934/cpaa.2012.11.1231

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Dynamics of non-autonomous nonclassical diffusion equations on $R^n$

Cung The Anh^1, and
Tang Quoc Bao^2,

1.
Department of Mathematics, Hanoi National University of Education, 36 Xuan Thuy, Cau Giay, Hanoi
2.
Faculty of Applied Mathematics and Informatics, Hanoi University of Science and Technology, No 1 Dai Co Viet, Hai Ba Trung, Hanoi

Received: November 30, 2010

Revised: April 30, 2011

Published: April 2012

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Abstract

We consider the Cauchy problem for a non-autonomous nonclassical diffusion equation of the form $u_t-\varepsilon\Delta u_t - \Delta u+f(u)+\lambda u=g(t)$ on $R^n$. Under an arbitrary polynomial growth order of the nonlinearity $f$ and a suitable exponent growth of the external force $g$, using the method of tail-estimates and the asymptotic a priori estimate method, we prove the existence of an $(H^{1}(R^n) L^{p}(R^n), H^{1}(R^n) L^{p}(R^n))$ - pullback attractor $\hat{A}_{\varepsilon}$ for the process associated to the problem. We also prove the upper semicontinuity of $\{\hat{A}_{\varepsilon}: \varepsilon\in [0,1]\}$ at $\varepsilon = 0$.

Keywords:

Non-autonomous nonclassical diffusion equation,
weak solution,
pullback attractor,
upper semicontinuity,
method of tail estimates,
a priori,

unbounded domain.

Mathematics Subject Classification: 35B41, 35K57, 35D05, 35B30.

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