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doi: 10.3934/dcdss.2021033

A hyperbolic-elliptic-parabolic PDE model describing chemotactic E. Coli colonies

CNRS, Laboratoire de Mathématiques d'Orsay, Université Paris-Saclay, 91405 Orsay cedex, France

* Corresponding author: Pierre Roux

Cet article est dédié á la mémoire du Professeur Ezzedine Zahrouni

Received  November 2020 Revised  January 2021 Published  March 2021

We study a modified version of an initial-boundary value problem describing the formation of colony patterns of bacteria Escherichia Coli. The original system of three parabolic equations was studied numerically and analytically and gave insights into the underlying mechanisms of chemotaxis. We focus here on the parabolic-elliptic-parabolic approximation and the hyperbolic-elliptic-parabolic limiting system which describes the case of pure chemotactic movement without random diffusion. We first construct local-in-time solutions for the parabolic-elliptic-parabolic system. Then we prove uniform a priori estimates and we use them along with a compactness argument in order to construct local-in-time solutions for the hyperbolic-elliptic-parabolic limiting system. Finally, we prove that some initial conditions give rise to solutions which blow-up in finite time in the $ L^\infty $ norm in all space dimensions. This last result is true even in space dimension 1, which is not the case for the full parabolic or parabolic-elliptic Keller-Segel systems.

Citation: Danielle Hilhorst, Pierre Roux. A hyperbolic-elliptic-parabolic PDE model describing chemotactic E. Coli colonies. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021033
References:
[1]

S. AgmondA. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I, Comm. Pure Appl. Math., 12 (1959), 623-727.  doi: 10.1002/cpa.3160120405.  Google Scholar

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P. Biler, Singularities of Solutions to Chemotaxis Systems, De Gruyter Series in Mathematics and Life Sciences, 6, De Gruyter, 2020.  Google Scholar

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A. BonamiD. HilhorstE. Logak and M. Mimura, A free boundary problem arising in a chemotaxis model, Free Boundary Problems, Theory and Applications, Pitman Res. Notes Math. Ser., 363 (1996), 368-373.   Google Scholar

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A. BonamiD. HilhorstE. Logak and M. Mimura, Singular limit of a chemotaxis-growth model, Adv. Differentials Equations, 6 (2001), 1173-1218.   Google Scholar

[7]

F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models. Applied Mathematical Sciences, New York: Springer, 2013. doi: 10.1007/978-1-4614-5975-0.  Google Scholar

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H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Verlag New York, 2011.  Google Scholar

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E. O. Budrene and H. C. Berg, Complex patterns formed by motile cells of Escherichia coli, Nature, 349 (1991), 630-633.  doi: 10.1038/349630a0.  Google Scholar

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E. O. Budrene and H. C. Berg, Dynamics of formation of symmetrical patterns by chemotactic bacteria, Nature, 376 (1995), 49-53.  doi: 10.1038/376049a0.  Google Scholar

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X. Cao, Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discrete Contin. Dyn. Syst., 35 (2015), 1891-1904.  doi: 10.3934/dcds.2015.35.1891.  Google Scholar

[12]

R. CelińskiD. HilhorstG. KarchM. Mimura and P. Roux, Mathematical treatment of PDE model of chemotactic E. coli colonies, J. Differential Equations, 278 (2021), 73-99.  doi: 10.1016/j.jde.2020.12.020.  Google Scholar

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E. FeireislD. HilhorstM. Mimura and R. Weidenfeld, On a nonlinear diffusion system with resource-consumer interaction, Hiroshima Math. J., 33 (2003), 253-295.  doi: 10.32917/hmj/1150997949.  Google Scholar

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X. FuQ. Griette and P. Magal, A cell-cell repulsion model on a hyperbolic Keller-Segel equation, J. Math. Biol., 80 (2020), 2257-2300.  doi: 10.1007/s00285-020-01495-w.  Google Scholar

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M. FunakiM. Mimura and T. Tsujikawa, Travelling front solutions arising in a chemotaxis-growth model, RIMS Kokyuroku, 1135 (2000), 52-76.   Google Scholar

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D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, second edition, 2001.  Google Scholar

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M. HenryD. Hilhorst and R. Schätzle, Convergence to a viscosity solution for an advection-reaction-diffusion equation arising from a chemotaxis-growth model, Hiroshima Math. J., 29 (1999), 591-630.  doi: 10.32917/hmj/1206124856.  Google Scholar

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T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58, (2009), 183–217. doi: 10.1007/s00285-008-0201-3.  Google Scholar

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

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D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. II, Jahresber. Deutsch. Math.-Verein., 106 (2004), 51-69.   Google Scholar

[22]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022.  Google Scholar

[23]

K. P. P. HtooM. Mimura and I. Takagi, Global solutions to a one-dimensional nonlinear parabolic system modeling colonial formation by chemotactic bacteria, Adv. Stud. Pure Math., 47 (2007), 613-622.  doi: 10.2969/aspm/04720613.  Google Scholar

[24]

K. Kang and A. Stevens, Blow-up and global solutions in a chemotaxis-growth system, Nonlinear Analysis, 135 (2016), 57-72.  doi: 10.1016/j.na.2016.01.017.  Google Scholar

[25]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[26]

J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differential Equations, 258 (2015), 1158-1191.  doi: 10.1016/j.jde.2014.10.016.  Google Scholar

[27]

J. Lankeit, Long-term behaviour in a chemotaxis-fluid system with logistic source, Math. Models Methods Appl. Sci., 26 (2016), 2071-2109.  doi: 10.1142/S021820251640008X.  Google Scholar

[28]

J. Lankeit, Chemotaxis can prevent thresholds on population density, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1499-1527.  doi: 10.3934/dcdsb.2015.20.1499.  Google Scholar

[29]

H. Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations, Publ. Res. Inst. Math. Sci., 15 (1979), 401-454.  doi: 10.2977/prims/1195188180.  Google Scholar

[30]

M. Mimura and T. Tsujikawa, Aggregating pattern dynamics in a chemotaxis model including growth, Phys. A, 230 (1996), 499-543.   Google Scholar

[31]

M. Mizukami and T. Yokota, A unified method for boundedness in fully parabolic chemotaxis systems with signal-dependent sensitivity, Math. Nachr., 290 (2017), 2648-2660.  doi: 10.1002/mana.201600399.  Google Scholar

[32]

L. Moonens, Private Communication. Google Scholar

[33]

T. Ogawa, Private Communication. Google Scholar

[34]

T. Ogawa and Y. Taniuchi, On blow-up criteria of smooth solutions to the 3-D Euler equations in a bounded domain, J. Differential Equations, 190 (2003), 39-63.  doi: 10.1016/S0022-0396(03)00013-5.  Google Scholar

[35]

K. OsakiT. TsujikawaA. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal., 51 (2002), 119-144.  doi: 10.1016/S0362-546X(01)00815-X.  Google Scholar

[36]

B. Perthame and A.-L. Dalibard, Existence of solutions of the hyperbolic Keller-Segel model, Trans. Amer. Math. Soc., 361 (2009), 2319-2335.  doi: 10.1090/S0002-9947-08-04656-4.  Google Scholar

[37]

E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, With the assistance of Timothy S. Murphy. Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, Ⅲ. Princeton University Press, Princeton, NJ, 1993.  Google Scholar

[38]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.  doi: 10.1080/03605300701319003.  Google Scholar

[39]

M. Winkler, How far can chemotactic cross-diffusion enforce exceeding carrying capacities?, J. Nonlinear Sci., 24 (2014), 809-855.  doi: 10.1007/s00332-014-9205-x.  Google Scholar

[40]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008.  Google Scholar

[41]

M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, J. Math. Anal. Appl., 384 (2011), 261-272.  doi: 10.1016/j.jmaa.2011.05.057.  Google Scholar

show all references

References:
[1]

S. AgmondA. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I, Comm. Pure Appl. Math., 12 (1959), 623-727.  doi: 10.1002/cpa.3160120405.  Google Scholar

[2]

A. AotaniM. Mimura and T. Mollee, A model aided understanding of spot pattern formation in chemotactic E. coli colonies, Jpn. J. Ind. Appl. Math., 27 (2010), 5-22.  doi: 10.1007/s13160-010-0011-z.  Google Scholar

[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.  Google Scholar

[4]

P. Biler, Singularities of Solutions to Chemotaxis Systems, De Gruyter Series in Mathematics and Life Sciences, 6, De Gruyter, 2020.  Google Scholar

[5]

A. BonamiD. HilhorstE. Logak and M. Mimura, A free boundary problem arising in a chemotaxis model, Free Boundary Problems, Theory and Applications, Pitman Res. Notes Math. Ser., 363 (1996), 368-373.   Google Scholar

[6]

A. BonamiD. HilhorstE. Logak and M. Mimura, Singular limit of a chemotaxis-growth model, Adv. Differentials Equations, 6 (2001), 1173-1218.   Google Scholar

[7]

F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models. Applied Mathematical Sciences, New York: Springer, 2013. doi: 10.1007/978-1-4614-5975-0.  Google Scholar

[8]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Verlag New York, 2011.  Google Scholar

[9]

E. O. Budrene and H. C. Berg, Complex patterns formed by motile cells of Escherichia coli, Nature, 349 (1991), 630-633.  doi: 10.1038/349630a0.  Google Scholar

[10]

E. O. Budrene and H. C. Berg, Dynamics of formation of symmetrical patterns by chemotactic bacteria, Nature, 376 (1995), 49-53.  doi: 10.1038/376049a0.  Google Scholar

[11]

X. Cao, Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discrete Contin. Dyn. Syst., 35 (2015), 1891-1904.  doi: 10.3934/dcds.2015.35.1891.  Google Scholar

[12]

R. CelińskiD. HilhorstG. KarchM. Mimura and P. Roux, Mathematical treatment of PDE model of chemotactic E. coli colonies, J. Differential Equations, 278 (2021), 73-99.  doi: 10.1016/j.jde.2020.12.020.  Google Scholar

[13]

X. Chen, Generation and propagation of interfaces in reaction-diffusion systems, Trans. Amer. Math. Soc., 334 (1992), 877-913.  doi: 10.1090/S0002-9947-1992-1144013-3.  Google Scholar

[14]

E. FeireislD. HilhorstM. Mimura and R. Weidenfeld, On a nonlinear diffusion system with resource-consumer interaction, Hiroshima Math. J., 33 (2003), 253-295.  doi: 10.32917/hmj/1150997949.  Google Scholar

[15]

X. FuQ. Griette and P. Magal, A cell-cell repulsion model on a hyperbolic Keller-Segel equation, J. Math. Biol., 80 (2020), 2257-2300.  doi: 10.1007/s00285-020-01495-w.  Google Scholar

[16]

M. FunakiM. Mimura and T. Tsujikawa, Travelling front solutions arising in a chemotaxis-growth model, RIMS Kokyuroku, 1135 (2000), 52-76.   Google Scholar

[17]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, second edition, 2001.  Google Scholar

[18]

M. HenryD. Hilhorst and R. Schätzle, Convergence to a viscosity solution for an advection-reaction-diffusion equation arising from a chemotaxis-growth model, Hiroshima Math. J., 29 (1999), 591-630.  doi: 10.32917/hmj/1206124856.  Google Scholar

[19]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58, (2009), 183–217. doi: 10.1007/s00285-008-0201-3.  Google Scholar

[20]

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

[21]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. II, Jahresber. Deutsch. Math.-Verein., 106 (2004), 51-69.   Google Scholar

[22]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022.  Google Scholar

[23]

K. P. P. HtooM. Mimura and I. Takagi, Global solutions to a one-dimensional nonlinear parabolic system modeling colonial formation by chemotactic bacteria, Adv. Stud. Pure Math., 47 (2007), 613-622.  doi: 10.2969/aspm/04720613.  Google Scholar

[24]

K. Kang and A. Stevens, Blow-up and global solutions in a chemotaxis-growth system, Nonlinear Analysis, 135 (2016), 57-72.  doi: 10.1016/j.na.2016.01.017.  Google Scholar

[25]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[26]

J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differential Equations, 258 (2015), 1158-1191.  doi: 10.1016/j.jde.2014.10.016.  Google Scholar

[27]

J. Lankeit, Long-term behaviour in a chemotaxis-fluid system with logistic source, Math. Models Methods Appl. Sci., 26 (2016), 2071-2109.  doi: 10.1142/S021820251640008X.  Google Scholar

[28]

J. Lankeit, Chemotaxis can prevent thresholds on population density, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1499-1527.  doi: 10.3934/dcdsb.2015.20.1499.  Google Scholar

[29]

H. Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations, Publ. Res. Inst. Math. Sci., 15 (1979), 401-454.  doi: 10.2977/prims/1195188180.  Google Scholar

[30]

M. Mimura and T. Tsujikawa, Aggregating pattern dynamics in a chemotaxis model including growth, Phys. A, 230 (1996), 499-543.   Google Scholar

[31]

M. Mizukami and T. Yokota, A unified method for boundedness in fully parabolic chemotaxis systems with signal-dependent sensitivity, Math. Nachr., 290 (2017), 2648-2660.  doi: 10.1002/mana.201600399.  Google Scholar

[32]

L. Moonens, Private Communication. Google Scholar

[33]

T. Ogawa, Private Communication. Google Scholar

[34]

T. Ogawa and Y. Taniuchi, On blow-up criteria of smooth solutions to the 3-D Euler equations in a bounded domain, J. Differential Equations, 190 (2003), 39-63.  doi: 10.1016/S0022-0396(03)00013-5.  Google Scholar

[35]

K. OsakiT. TsujikawaA. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal., 51 (2002), 119-144.  doi: 10.1016/S0362-546X(01)00815-X.  Google Scholar

[36]

B. Perthame and A.-L. Dalibard, Existence of solutions of the hyperbolic Keller-Segel model, Trans. Amer. Math. Soc., 361 (2009), 2319-2335.  doi: 10.1090/S0002-9947-08-04656-4.  Google Scholar

[37]

E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, With the assistance of Timothy S. Murphy. Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, Ⅲ. Princeton University Press, Princeton, NJ, 1993.  Google Scholar

[38]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.  doi: 10.1080/03605300701319003.  Google Scholar

[39]

M. Winkler, How far can chemotactic cross-diffusion enforce exceeding carrying capacities?, J. Nonlinear Sci., 24 (2014), 809-855.  doi: 10.1007/s00332-014-9205-x.  Google Scholar

[40]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008.  Google Scholar

[41]

M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, J. Math. Anal. Appl., 384 (2011), 261-272.  doi: 10.1016/j.jmaa.2011.05.057.  Google Scholar

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