IJMNTA  Vol.3 No.1 , March 2014
Exponential Attractors of the Nonclassical Diffusion Equations with Lower Regular Forcing Term
ABSTRACT

In this article, we prove the existence of exponential attractors of the nonclassical diffusion equation with critical nonlinearity and lower regular forcing term. As an additional product, we show that the fractal dimension of the global attractors of this problem is finite.


Cite this paper
Zhang, Y. and Ma, Q. (2014) Exponential Attractors of the Nonclassical Diffusion Equations with Lower Regular Forcing Term. International Journal of Modern Nonlinear Theory and Application, 3, 15-22. doi: 10.4236/ijmnta.2014.31003.
References
[1]   Aifantis, E.C. (1980) On the Problem of Diffusion in Solids. Acta Mechanica, 37, 265-296.
http://dx.doi.org/10.1007/BF01202949

[2]   Kuttler, K. and Aifantis, E. (1988) Quasilinear Evolution Equations in Nonclassical Diffusion. SIAM Journal on Applied Mathematics, 19, 110-120. http://dx.doi.org/10.1137/0519008

[3]   Peter, J.G. and Gurtin, M.E. (1968) On the Theory of Heat Condition Involving Two Temperatures. Zeitschrift für angewandte Mathematik und Physik ZAMP, 19, 614-627. http://dx.doi.org/10.1007/BF01594969

[4]   Xiao, Y. (2002) Attractors for a Nonclassical Diffusion. Acta Mathematicae Applicatae Sinica, 18, 273-276.
http://dx.doi.org/10.1007/s102550200026

[5]   Wang, S., Li, D. and Zhong, C. (2006) On the Dynamics of a Class of Nonclassical Parabolic Equation. Journal of Mathematical Analysis and Applications, 317, 565-582. http://dx.doi.org/10.1016/j.jmaa.2005.06.094

[6]   Sun, C., Wang, S. and Zhong, C. (2007) Global Attractors for a Nonclassical Diffusion Equation. Acta Mathematicae Applicatae Sinica, 23, 1271-1280.

[7]   Sun, C. and Yang, M. (2008) Dynamics of the Nonclassical Diffusion Equations. Asymptotic Analysis, 59, 51-81.

[8]   Sun, C. (2009) Asympotic Regularity for Some Dissipative Equations. Journal of Differential Equations, 248, 342-362. http://dx.doi.org/10.1016/j.jde.2009.08.007

[9]   Wu, H. and Zhang, Z. (2011) Asymptotic Regularity for the Nonclassical Diffusion Equation with Lower Regular Foring Term. Dynamical Systems, 26, 391-400. http://dx.doi.org/10.1080/14689367.2011.562185

[10]   Sun, C., Yang, L. and Duan, J. (2011) Asymptotic Behavior for a Semilinear Second Order Evolution Equation. Transactions of the American Mathematical Society, 363, 6085-6109.
http://dx.doi.org/10.1090/S0002-9947-2011-05373-0

[11]   Efendiev, M., Miranville, A. and Zelik, S.V. (2005) Exponential Attractors and Finite-Dimensional Reduction of NonAutonomous Dynamical Systems. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 135, 703-730.

[12]   Miranville, A. and Zelik, S.V. (2008) Attracts for Dissipative Partial Different Equations in Bounded and Unbounded Domains, In: Dafermos, C.M. and Pokorny, M., Eds., Evolutionary Equations, Elsevier, Amesterdam, 103.

[13]   Fabrie, P., Galusinski, C., Miranville, A. and Zelik, S. (2004) Uniform Exponential Attractors for a Singularly Perturbed Damped Wave Equation. Discrete and Continuous Dynamical Systems, 10, 211-238.
http://dx.doi.org/10.3934/dcds.2004.10.211

[14]   Pata, V. and Squassina, M. (2005) On the Strongly Damped Wave Equation. Communications in Mathematical Physics, 253, 511-533. http://dx.doi.org/10.1007/s00220-004-1233-1

 
 
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