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February 2018, 38(2): 651-673. doi: 10.3934/dcds.2018028

On an exponential attractor for a class of PDEs with degenerate diffusion and chemotaxis

1. 

Institute of Computational Biology, Helmholtz Zentrum München, Ingolstädter Landstr. 1,85764 Neuherberg, Germany

2. 

Felix-Klein-Zentrum für Mathematik, Technische Universität Kaiserslautern, Paul-Ehrlich-Str. 31,67663 Kaiserslautern, Germany

* Corresponding author: Anna Zhigun

Received  May 2017 Revised  September 2017 Published  February 2018

In this article we deal with a class of strongly coupled parabolic systems that encompasses two different effects: degenerate diffusion and chemotaxis. Such classes of equations arise in the mesoscale level modeling of biomass spreading mechanisms via chemotaxis. We show the existence of an exponential attractor and, hence, of a finite-dimensional global attractor under certain 'balance conditions' on the order of the degeneracy and the growth of the chemotactic function.

Citation: Messoud Efendiev, Anna Zhigun. On an exponential attractor for a class of PDEs with degenerate diffusion and chemotaxis. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 651-673. doi: 10.3934/dcds.2018028
References:
[1]

M. AidaT. TsujikawaM. EfendievA. Yagi and M. Mimura, Lower estimate of the attractor dimension for a chemotaxis growth system, J. London Math. Soc. (2), 74 (2006), 453-474. doi: 10.1112/S0024610706023015.

[2]

D. G. Aronson and P. Bénilan, Régularité des solutions de l'équation des milieux poreux dans $\textbf{R}^{N}$, C. R. Acad. Sci. Paris Sér. A-B, 288 (1979), A103-A105.

[3]

N. BellomoA. BellouquidY. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763. doi: 10.1142/S021820251550044X.

[4]

L. A. Caffarelli and A. Friedman, Regularity of the free boundary of a gas flow in an n-dimensional porous medium, Indiana Univ. Math. J., 29 (1980), 361-391. doi: 10.1512/iumj.1980.29.29027.

[5]

E. DiBenedetto, Continuity of weak solutions to a general porous medium equation, Indiana Univ. Math. J., 32 (1983), 83-118. doi: 10.1512/iumj.1983.32.32008.

[6]

H. EberlE. JalbertA. Dumitrache and G. Wolfaardt, A spatially explicit model of inverse colony formation of cellulolytic biofilms, Biochemical Engineering Journal, 122 (2017), 141-151.

[7]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations vol. 37 of RAM: Research in Applied Mathematics, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994.

[8]

M. Efendiev, Finite and Infinite Dimensional Attractors for Evolution Equations of Mathematical Physics vol. 33 of GAKUTO International Series. Mathematical Sciences and Applications, Gakkōtosho Co., Ltd., Tokyo, 2010.

[9]

M. Efendiev, Attractors for Degenerate Parabolic Type Equations vol. 192 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI; Real Sociedad Matemática Española, Madrid, 2013. doi: 10.1090/surv/192.

[10]

M. EfendievE. Nakaguchi and K. Osaki, Dimension estimate of the exponential attractor for the chemotaxis-growth system, Glasg. Math. J., 50 (2008), 483-497. doi: 10.1017/S0017089508004357.

[11]

M. EfendievE. Nakaguchi and W. L. Wendland, Uniform estimate of dimension of the global attractor for a semi-discretized chemotaxis-growth system, Discrete Contin. Dyn. Syst., (2007), 334-343.

[12]

M. EfendievE. Nakaguchi and W. L. Wendland, Dimension estimate of the global attractor for a semi-discretized chemotaxis-growth system by conservative upwind finite-element scheme, J. Math. Anal. Appl., 358 (2009), 136-147. doi: 10.1016/j.jmaa.2009.04.025.

[13]

M. Efendiev and M. Ôtani, Infinite-dimensional attractors for evolution equations with p-Laplacian and their {K}olmogorov entropy, Differential Integral Equations, 20 (2007), 1201-1209.

[14]

M. Efendiev and M. Ôtani, Infinite-dimensional attractors for parabolic equations with p-Laplacian in heterogeneous medium, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 565-582. doi: 10.1016/j.anihpc.2011.03.006.

[15]

M. Efendiev and T. Senba, On the well posedness of a class of PDEs including porous medium and chemotaxis effect, Adv. Differential Equations, 16 (2011), 937-954.

[16]

M. Efendiev and A. Yagi, Continuous dependence on a parameter of exponential attractors for chemotaxis-growth system, J. Math. Soc. Japan, 57 (2005), 167-181, URL http: //projecteuclid.org/euclid.jmsj/1160745820.

[17]

M. Efendiev and S. Zelik, Finite-and infinite-dimensional attractors for porous media equations, Proc. Lond. Math. Soc. (3), 96 (2008), 51-77. doi: 10.1112/plms/pdm026.

[18]

M. Efendiev and A. Zhigun, On a 'balance' condition for a class of PDEs including porous medium and chemotaxis effect: non-autonomous case, Adv. Math. Sci. Appl., 21 (2011), 285-304.

[19]

M. Efendiev and A. Zhigun, On a global uniform pull-back attractor of a class of PDEs with degenerate diffusion and chemotaxis, Adv. Math. Sci. Appl., 23 (2013), 437-460.

[20]

M. Efendiev and A. Zhigun, On a global uniform pullback attractor of a class of PDEs with degenerate diffusion and chemotaxis in one dimension, in Recent trends in dynamical systems, vol. 35 of Springer Proc. Math. Stat., Springer, Basel, 2013,179-203. doi: 10.1007/978-3-0348-0451-6_9.

[21]

M. EfendievA. Zhigun and T. Senba, On a weak attractor of a class of PDEs with degenerate diffusion and chemotaxis, J. Math. Soc. Japan, 66 (2014), 1133-1153. doi: 10.2969/jmsj/06641133.

[22]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165.

[23]

A. V. Ivanov, Estimates for the Hölder constant of generalized solutions of degenerate parabolic equations, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 152 (1986), 21-44,181. doi: 10.1007/BF01094184.

[24]

O. Ladyzhenskaya, V. Solonnikov and N. Ural'tseva, Linear and Quasi-Linear Equations of Parabolic Type, (Russian) Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23 American Mathematical Society, Providence, R. I., 1968.

[25]

E. Nakaguchi, M. Efendiev and W. L. Wendland, Comparison of approximation schemes for chemotaxis-growth system via dimensions of global attractors, in Nonlinear phenomena with energy dissipation, vol. 29 of GAKUTO Internat. Ser. Math. Sci. Appl., Gakkōtosho, Tokyo, 2008,305-312.

[26]

E. Nakaguchi and K. Osaki, Global solutions and exponential attractors of a parabolic-parabolic system for chemotaxis with subquadratic degradation, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2627-2646. doi: 10.3934/dcdsb.2013.18.2627.

[27]

J. L. Vázquez, The Porous Medium Equation Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007, Mathematical theory.

[28]

A. Zhigun, Biofilm Models with Various Nonlinear Effects: Long-time Behavior of Solutions Dissertation, Technische Universität München, München, 2013, URL https://mediatum.ub.tum.de/node?id=1129649.

[29]

W. P. Ziemer, Interior and boundary continuity of weak solutions of degenerate parabolic equations, Trans. Amer. Math. Soc., 271 (1982), 733-748. doi: 10.2307/1998907.

show all references

References:
[1]

M. AidaT. TsujikawaM. EfendievA. Yagi and M. Mimura, Lower estimate of the attractor dimension for a chemotaxis growth system, J. London Math. Soc. (2), 74 (2006), 453-474. doi: 10.1112/S0024610706023015.

[2]

D. G. Aronson and P. Bénilan, Régularité des solutions de l'équation des milieux poreux dans $\textbf{R}^{N}$, C. R. Acad. Sci. Paris Sér. A-B, 288 (1979), A103-A105.

[3]

N. BellomoA. BellouquidY. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763. doi: 10.1142/S021820251550044X.

[4]

L. A. Caffarelli and A. Friedman, Regularity of the free boundary of a gas flow in an n-dimensional porous medium, Indiana Univ. Math. J., 29 (1980), 361-391. doi: 10.1512/iumj.1980.29.29027.

[5]

E. DiBenedetto, Continuity of weak solutions to a general porous medium equation, Indiana Univ. Math. J., 32 (1983), 83-118. doi: 10.1512/iumj.1983.32.32008.

[6]

H. EberlE. JalbertA. Dumitrache and G. Wolfaardt, A spatially explicit model of inverse colony formation of cellulolytic biofilms, Biochemical Engineering Journal, 122 (2017), 141-151.

[7]

A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations vol. 37 of RAM: Research in Applied Mathematics, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994.

[8]

M. Efendiev, Finite and Infinite Dimensional Attractors for Evolution Equations of Mathematical Physics vol. 33 of GAKUTO International Series. Mathematical Sciences and Applications, Gakkōtosho Co., Ltd., Tokyo, 2010.

[9]

M. Efendiev, Attractors for Degenerate Parabolic Type Equations vol. 192 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI; Real Sociedad Matemática Española, Madrid, 2013. doi: 10.1090/surv/192.

[10]

M. EfendievE. Nakaguchi and K. Osaki, Dimension estimate of the exponential attractor for the chemotaxis-growth system, Glasg. Math. J., 50 (2008), 483-497. doi: 10.1017/S0017089508004357.

[11]

M. EfendievE. Nakaguchi and W. L. Wendland, Uniform estimate of dimension of the global attractor for a semi-discretized chemotaxis-growth system, Discrete Contin. Dyn. Syst., (2007), 334-343.

[12]

M. EfendievE. Nakaguchi and W. L. Wendland, Dimension estimate of the global attractor for a semi-discretized chemotaxis-growth system by conservative upwind finite-element scheme, J. Math. Anal. Appl., 358 (2009), 136-147. doi: 10.1016/j.jmaa.2009.04.025.

[13]

M. Efendiev and M. Ôtani, Infinite-dimensional attractors for evolution equations with p-Laplacian and their {K}olmogorov entropy, Differential Integral Equations, 20 (2007), 1201-1209.

[14]

M. Efendiev and M. Ôtani, Infinite-dimensional attractors for parabolic equations with p-Laplacian in heterogeneous medium, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 565-582. doi: 10.1016/j.anihpc.2011.03.006.

[15]

M. Efendiev and T. Senba, On the well posedness of a class of PDEs including porous medium and chemotaxis effect, Adv. Differential Equations, 16 (2011), 937-954.

[16]

M. Efendiev and A. Yagi, Continuous dependence on a parameter of exponential attractors for chemotaxis-growth system, J. Math. Soc. Japan, 57 (2005), 167-181, URL http: //projecteuclid.org/euclid.jmsj/1160745820.

[17]

M. Efendiev and S. Zelik, Finite-and infinite-dimensional attractors for porous media equations, Proc. Lond. Math. Soc. (3), 96 (2008), 51-77. doi: 10.1112/plms/pdm026.

[18]

M. Efendiev and A. Zhigun, On a 'balance' condition for a class of PDEs including porous medium and chemotaxis effect: non-autonomous case, Adv. Math. Sci. Appl., 21 (2011), 285-304.

[19]

M. Efendiev and A. Zhigun, On a global uniform pull-back attractor of a class of PDEs with degenerate diffusion and chemotaxis, Adv. Math. Sci. Appl., 23 (2013), 437-460.

[20]

M. Efendiev and A. Zhigun, On a global uniform pullback attractor of a class of PDEs with degenerate diffusion and chemotaxis in one dimension, in Recent trends in dynamical systems, vol. 35 of Springer Proc. Math. Stat., Springer, Basel, 2013,179-203. doi: 10.1007/978-3-0348-0451-6_9.

[21]

M. EfendievA. Zhigun and T. Senba, On a weak attractor of a class of PDEs with degenerate diffusion and chemotaxis, J. Math. Soc. Japan, 66 (2014), 1133-1153. doi: 10.2969/jmsj/06641133.

[22]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165.

[23]

A. V. Ivanov, Estimates for the Hölder constant of generalized solutions of degenerate parabolic equations, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 152 (1986), 21-44,181. doi: 10.1007/BF01094184.

[24]

O. Ladyzhenskaya, V. Solonnikov and N. Ural'tseva, Linear and Quasi-Linear Equations of Parabolic Type, (Russian) Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23 American Mathematical Society, Providence, R. I., 1968.

[25]

E. Nakaguchi, M. Efendiev and W. L. Wendland, Comparison of approximation schemes for chemotaxis-growth system via dimensions of global attractors, in Nonlinear phenomena with energy dissipation, vol. 29 of GAKUTO Internat. Ser. Math. Sci. Appl., Gakkōtosho, Tokyo, 2008,305-312.

[26]

E. Nakaguchi and K. Osaki, Global solutions and exponential attractors of a parabolic-parabolic system for chemotaxis with subquadratic degradation, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2627-2646. doi: 10.3934/dcdsb.2013.18.2627.

[27]

J. L. Vázquez, The Porous Medium Equation Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007, Mathematical theory.

[28]

A. Zhigun, Biofilm Models with Various Nonlinear Effects: Long-time Behavior of Solutions Dissertation, Technische Universität München, München, 2013, URL https://mediatum.ub.tum.de/node?id=1129649.

[29]

W. P. Ziemer, Interior and boundary continuity of weak solutions of degenerate parabolic equations, Trans. Amer. Math. Soc., 271 (1982), 733-748. doi: 10.2307/1998907.

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