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Exponential attractors for abstract parabolic systems with bounded delay

Published online by Cambridge University Press:  17 April 2009

Dalibor Pražák
Dalibor Pražák
Affiliation:
Department of Mathematical Analysis, Charles University, Sokolovská83CZ-18675 Prague 8, Czech Republic e-mail: prazak@karlin.mff.cuni.cz
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We show that a suitable adaptation of the so-called method of trajectories can be used to construct an exponential attractor for a very general class of nonlinear reaction-diffusion systems with a bounded delay.

In particular, we assume that the dependence on the past history is controlled via convolution with a possibly singular measure. Assuming a priori that the solutions are bounded, a simple proof of the existence of an exponential attractor is given under very little regularity requirements.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 76 , Issue 2 , October 2007 , pp. 285 - 295
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Chepyzhov, V.V., Gatti, S., Grasselli, M., Miranville, A. and Pata, V., ‘Trajectory and global attractors for evolution equations with memory’, Appl. Math. Lett. 19 (2006), 87–96.CrossRefGoogle Scholar
[2]Chepyzhov, V.V. and Miranville, A., ‘On trajectory and global attractors for semilinear heat equations with fading memory’, Indiana Univ. Math. J. 55 (2006), 119–167.Google Scholar
[3]Chepyzhov, V.V. and Vishik, M.I., Attractors for equations of mathematical physics, Amer. Math. Soc. Colloq. Publ. 49 (American Mathematical Society, Providence, R.I., 2002).Google Scholar
[4]Chepyzhov, V.V., Vladimir, V. and Vishik, M.I., ‘Evolution equations and their trajectory attractors’, J. Math. Pures Appl. 76 (1997), 913–964.Google Scholar
[5]Eden, A., Foias, C., Nicolaenko, B. and Temam, R., Exponential attractors for dissipative evolution equations, Research in Applied Mathematics 37 (Masson, Paris, 1994).Google Scholar
[6]Efendiev, M., Miranville, A. and Zelik, S., ‘Exponential attractors for a nonlinear reaction-diffusion system in R3’, C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), 713–718.CrossRefGoogle Scholar
[7]Feireisl, E., ‘Propagation of oscillations, complete trajectories and attractors for compressible flows’, NoDEA Nonlinear Differential Equations Appl. 10 (2003), 33–55.Google Scholar
[8]Friesecke, G., ‘Exponentially growing solutions for a delay-diffusion equation with negative feedback’, J. Differential Equations 98 (1992), 1–18.Google Scholar
[9]Grasselli, M. and Pata, V., ‘Uniform attractors of nonautonomous dynamical systems with memory’, in Progr. Nonlinear Differential Equations Appl. 50 (Birkhäuser, Basel, 2002), pp. 155–178.Google Scholar
[10]Grasselli, M. and PraŽák, D., ‘Exponential attractors for a class of reaction-diffusion problems with time delays’, J. Evol. Equ. (to appear).Google Scholar
[11]Luckhaus, S., ‘Global boundedness for a delay-differential equation’, Trans. Amer. Math. Soc. 294 (1986), 767–774.Google Scholar
[12]Málek, J. and Nečas, J., ‘A finite-dimensional attractor for three-dimensional flow of incompressible fluids’, J. Differential Equations 127 (1996), 498–518.CrossRefGoogle Scholar
[13]Málek, J. and PraŽák, D., ‘Large time behavior via the method of l-trajectories’, J. Differential Equations 181 (2002), 243–279.Google Scholar
[14]Pao, C.V., ‘Dynamics of nonlinear parabolic systems with time delays’, J. Math. Anal. Appl. 198 (1996), 751–779.CrossRefGoogle Scholar
[15]Pao, C.V., ‘Systems of parabolic equations with continuous and discrete delays’, J. Math. Anal. Appl. 205 (1997), 157–185.CrossRefGoogle Scholar
[16]Pao, C.V., ‘Stability and attractivity of periodic solutions of parabolic systems with time delays’, J. Math. Anal. Appl. 304 (2005), 423–450.CrossRefGoogle Scholar
[17]PraŽák, D., ‘Exponential attractor for the delayed logistic equation with a nonlinear diffusion’, Discrete Contin. Dyn. Syst. (supplement) (2003), 717–726.Google Scholar
[18]PraŽák, D., ‘A necessary and sufficient condition for the existence of an exponential attractor’, Cent. Eur. J. Math. 1 (2003), 411–417.CrossRefGoogle Scholar
[19]PraŽák, D., ‘On the dynamics of equations with infinite delay’, Cent. Eur. J. Math, (electronic) 4 (2006), 635–647.CrossRefGoogle Scholar
[20]Rezounenko, A.V., ‘Partial differential equations with discrete and distributed state-dependent delays’, J. Math. Anal. Appl. 326 (2007), 1031–1045.CrossRefGoogle Scholar
[21]Robinson, J.C., Infinite-dimensional dynamical systems, Cambridge Texts in Applied Mathematics (Cambridge University Press, Cambridge, 2001).CrossRefGoogle Scholar
[22]Sell, G.R., ‘Global attractors for the three-dimensional Navier-Stokes equations’, J. Dynam. Differential Equations 8 (1996), 1–33.CrossRefGoogle Scholar
[23]Wang, L. and Xu, D., ‘Asymptotic behavior of a class of reaction-diffusion equations with delays’, J. Math. Anal. Appl. 281 (2003), 439–453.Google Scholar