August  2021, 14(8): 2993-3015. 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  August 2021 Early access  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 and Continuous Dynamical Systems - S, 2021, 14 (8) : 2993-3015. 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.

[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.

[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]

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

[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. 

[6]

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

[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.

[8]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

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

[17]

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

[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.

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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.

[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. 

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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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[30]

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

[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.

[32]

L. Moonens, Private Communication.

[33]

T. Ogawa, Private Communication.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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]

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

[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. 

[6]

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

[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.

[8]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

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

[17]

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

[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.

[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.

[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. 

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[30]

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

[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.

[32]

L. Moonens, Private Communication.

[33]

T. Ogawa, Private Communication.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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