Abstract
In this chapter, we deal with a class of nonautonomous degenerate parabolic systems that encompasses two different effects: porous medium and chemotaxis. Such classes of equations arise in the mesoscale level modeling of biomass spreading mechanisms via chemotaxis. Under certain “balance” conditions on the order of the porous medium degeneracy and the growth of the chemotactic function, we establish the existence of a strong uniform pull back attractor for the case of one spatial dimension, thus improving our previous study, where a weak attractor was constructed.
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References
Adams, R.A.: Sobolev Spaces. Academic, New York (1975)
Babin, A.V., Vishik, M.I.: Attractors of Evolution Equations. North-Holland, Amsterdam (1992)
Efendiev, M.A.: Attractors of Degenerate Parabolic Type Equations. AMS (2013, in press)
Efendiev, M.A.: Finite and Infinite Dimensional Attractors for Evolution Equations of Mathematical Physics, vol. 33. Gakkotoscho International Series, Tokyo (2010)
Efendiev, M.A., Yamamoto, Y., Yagi, A.: Exponential attractors for non-autonomous dissipative system. J. Math. Soc. Japan 63(2), 647–673 (2011)
Efendiev, M.A., Zhigun, A.: On a ‘balance’ condition for a class of PDEs including porous medium and chemotaxis effect: nonautonomous case. Adv. Math. Sci. Appl. 21, 285–304 (2011)
Efendiev, M.A., Zhigun, A., Senba, T.: On a weak attractor of a class of PDE with degenerate diffusion and chemotaxis (2013, in press)
Horstmann, D.: From 1970 until present: the Keller–Segel model in chemotaxis and its consequences, Part I. Jahresbericht der DMV 105(3), 103–165 (2003)
Ishida, S., Yokota, T.: Global existence of weak solutions to quasilinear degenerate Keller–Segel systems of parabolic-parabolic type. J. Differ. Equ. 252(2), 1421–1440 (2012)
Kloeden, P.: Pullback attractors of nonautonomous semidynamical systems. Stoch. Dyn. 3(1), 101–112 (2003)
Luckhaus, S., Sugiyama, Y.: Asymptotic profile with the optimal convergence rate for a parabolic equation of chemotaxis in super-critical cases. Indiana Univ. Math. J. 56(3), 1279–1297 (2007)
Luckhaus, S., Sugiyama, Y.: Large time behavior of solutions in super-critical cases to degenerate Keller–Segel systems. ESAIM, Math. Model. Numer. Anal. 40(3), 597–621 (2006)
Schmalfuss, B.: Attractors for the nonautonomous dynamical systems. In: Gröger, K., Fiedler, B., Sprekels, J. (eds.) Proceedings of EQUADIFF99. World Scientific (2000)
Sugiyama, Y.: Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenerate Keller–Segel system. Differ. Integral Equ. 20, 841–876 (2006)
Yagi, A.: Abstract Parabolic Evolution Equations and their Applications. Springer Monographs in Mathematics. Springer, Berlin (2010)
Acknowledgements
The authors express their thanks to T. Senba for many stimulating discussions.
Anna Zhigun is sponsored by The Elite Network of Bavaria.
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Efendiev, M., Zhigun, A. (2013). On a Global Uniform Pullback Attractor of a Class of PDEs with Degenerate Diffusion and Chemotaxis in One Dimension. In: Johann, A., Kruse, HP., Rupp, F., Schmitz, S. (eds) Recent Trends in Dynamical Systems. Springer Proceedings in Mathematics & Statistics, vol 35. Springer, Basel. https://doi.org/10.1007/978-3-0348-0451-6_9
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