On a Parabolic-ODE system of chemotaxis

Mihaela Negreanu; J. Ignacio Tello

doi:10.3934/dcdss.2020016

Previous Article
Existence of traveling wave solutions to parabolic-elliptic-elliptic chemotaxis systems with logistic source
DCDS-S Home
This Issue
Next Article
Improvement of conditions for asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity

February 2020, 13(2): 279-292. doi: 10.3934/dcdss.2020016

On a Parabolic-ODE system of chemotaxis

Mihaela Negreanu ^1, and J. Ignacio Tello ^2,3,,

1.	Departamento de Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain
2.	Departamento de Matemática Aplicada, E.T.S.I. Sistemas Informáticos, Universidad Politécnica de Madrid 28031, Spain
3.	Center for Computation and Simulation, Universidad Politécnica de Madrid, 28660 Boadilla del Monte, Madrid, Spain

^† Corresponding author

Received May 2017 Revised April 2018 Published January 2019

Fund Project: This work has been supported by the Project MTM2017-83391-P from MICINN (Spain).

In this article we consider a coupled system of differential equations to describe the evolution of a biological species. The system consists of two equations, a second order parabolic PDE of nonlinear type coupled to an ODE. The system contains chemotactic terms with constant chemotaxis coefficient describing the evolution of a biological species "

$u$

" which moves towards a higher concentration of a chemical species "

$v$

" in a bounded domain of

$ \mathbb{R}^n$

. The chemical "

$v$

" is assumed to be a non-diffusive substance or with neglectable diffusion properties, satisfying the equation

$v_t = h(u, v).$

We obtain results concerning the bifurcation of constant steady states under the assumption

$ h_v+χ u h_u>0 $

with growth terms

$g$

. The Parabolic-ODE problem is also considered for the case

$h_v+χ u h_u = 0$

without growth terms, i.e.

$g \equiv 0$

. Global existence of solutions is obtained for a range of initial data.

Keywords: Parabolic-ODE, chemotaxis, global existence.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

Citation: Mihaela Negreanu, J. Ignacio Tello. On a Parabolic-ODE system of chemotaxis. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 279-292. doi: 10.3934/dcdss.2020016

References:

[1]	A. R. A. Anderson and M. A. I. Chaplain, Continuous and discrete mathematical models of tumor-induced angiogenesis, Bull. Math. Biology, 60 (1998), 857-899. doi: 10.1006/bulm.1998.0042. Google Scholar
[2]	T. Black, Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals, Discrete& Continuous Dynamical Systems-Series B, 22 (2017), 1253-1272. doi: 10.3934/dcdsb.2017061. Google Scholar
[3]	T. Black, J. Lankeit and M. Mizukami, On the weakly competitive case in a two-species chemotaxis model, IMA J Appl Math., 81 (2016), 860-876. doi: 10.1093/imamat/hxw036. Google Scholar
[4]	T. Bollenbach, K. Kruse, P. Pantazis, M. González-Gaitán and F. Jülicher, Morphogen transport in Epithelia, Physical Review E Stat Nonlin Soft Matter Phys, 75 (2007), 011901. doi: 10.1103/PhysRevE.75.011901. Google Scholar
[5]	N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Mathematical Models and Methods in Applied Sciences, 25 (2015), 1663-1763. doi: 10.1142/S021820251550044X. Google Scholar
[6]	C. Conca and E. Espejo, Threshold condition for global existence and blow-up to a radially symmetric drift-diffusion system, Appl. Math. Lett., 25 (2012), 352-356. doi: 10.1016/j.aml.2011.09.013. Google Scholar
[7]	L. Edelstein-Keshet, Mathematical Models in Biology, Society for Industrial and Applied Mathematics, Philadelphia 2005. doi: 10.1137/1.9780898719147. Google Scholar
[8]	A. Fasano, A. Mancini and M. Primicerio, Equilibrium of two populations subject to chemotaxis, Math. Models Methods Appl. Sci., 14 (2004), 503-533. doi: 10.1142/S0218202504003337. Google Scholar
[9]	M. A. Fontelos, A. Friedman and B. Hu, Mathematical analysis of a model for the initiation of angiogenesis, SIAM Math. Anal., 33 (2002), 1330-1355. doi: 10.1137/S0036141001385046. Google Scholar
[10]	A. Friedman and J. I. Tello, Stability of solutions of chemotaxis equations in reinforced random walks, J. Math. Anal. Appl., 272 (2002), 138-163. doi: 10.1016/S0022-247X(02)00147-6. Google Scholar
[11]	M. Hirata, S. Kurima, M. Mizukami and T. Yokota, Boundedness and stabilization in a twodimensional two-species chemotaxis-Navier- Stokes system with competitive kinetics, Journal of Differential Equations, 263 (2017), 470-490. doi: 10.1016/j.jde.2017.02.045. Google Scholar
[12]	D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, Jahresber. Dtsch. Math. Ver., 105 (2003), 103-165. Google Scholar
[13]	D. Horstmann, Generalizing the Keller-Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, J. Nonlinear Sci., 21 (2011), 231-270. doi: 10.1007/s00332-010-9082-x. Google Scholar
[14]	E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5. Google Scholar
[15]	E. F. Keller and L. A. Segel, A model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6. Google Scholar
[16]	H. Kielhöfer, Bifurcation Theory, Springer, 2004. doi: 10.1007/b97365. Google Scholar
[17]	P. Krzyżanowski, P. Laurençot and D. Wrzosek, Mathematical models of receptor-mediated transport of morphogens, M3AS, 20 (2010), 2021-2052. doi: 10.1142/S0218202510004866. Google Scholar
[18]	A. Kubo, H. Hoshino and K. Kimura, Global existence and asymptotic behaviour of solutions for nonlinear evolution equations related to a tumour invasion, Proceedings of 10th AIMS Conference, (2015), 733-744. doi: 10.3934/proc.2015.0733. Google Scholar
[19]	A. Kubo and T. Suzuki, Mathematical models of tumour angiogenesis, Journal of Computational and Applied Mathematics, 204 (2007), 48-55. doi: 10.1016/j.cam.2006.04.027. Google Scholar
[20]	A. Kubo and J. I. Tello, Mathematical analysis of a model of chemotaxis with competition terms, Differential and Integral Equations, 29 (2016), 441-454. Google Scholar
[21]	D. A. Lauffenburger, Quantitative studies of bacterial chemotaxis and microbial population dynamics, Microb. Ecol., 22 (1991), 175-185. doi: 10.1007/BF02540222. Google Scholar
[22]	H. A. Levine and B. D. Sleeman, A system of reaction diffusion equation arising in the theory of reinforced random walks, SIAM J. Appl. Math., 57 (1997), 683-730. doi: 10.1137/S0036139995291106. Google Scholar
[23]	H. A. Levine and J. Renclawowicz, Singularity formation in chemotaxis - a conjecture of nagai, SIAM J. Appl. Math., 65 (2004), 336-360. doi: 10.1137/S0036139903431725. Google Scholar
[24]	H. A. Levine, B. P. Sleeman and N. Nilsen-Hamilton, A mathematical modeling for the roles of pericytes and macrophages in the initiation of angiogenesis I. The role of protease inhibitors in preventing angiogenesis, Mathematical Biosciences, 168 (2000), 75-115. doi: 10.1016/S0025-5564(00)00034-1. Google Scholar
[25]	Y. Li, K. Lin and C. Mu, Boundedness and asymptotic behavior of solutions to a chemotaxishaptotaxis model in high dimensions, Applied Mathematics Letters, 50 (2015), 91-97. doi: 10.1016/j.aml.2015.06.010. Google Scholar
[26]	M. Malogrosz, Well-posedness and asymptotic behavior of a multidimensonal model of morphogen transport, J. Evol. Eq., 12 (2012), 353-366. doi: 10.1007/s00028-012-0135-5. Google Scholar
[27]	M. Małogrosz, A model of morphogen transport in the presence of glypicans II, Journal of Mathematical Analysis and Applications, 433 (2016), 642-680. doi: 10.1016/j.jmaa.2015.07.053. Google Scholar
[28]	J. H. Merking, D. J. Needham and B. D. Sleeman, A mathematical model for the spread of morphogens with density dependent chemosensitivity, Nonlinearity, 18 (2005), 2745-2773. doi: 10.1088/0951-7715/18/6/018. Google Scholar
[29]	J. H. Merking and B. D. Sleeman, On the spread of morphogens, J. Math. Biol., 51 (2005), 1-17. doi: 10.1007/s00285-004-0308-0. Google Scholar
[30]	M. Mizukami and T. Yokota, Global existence and asymptotic stability of solutions to a twospecies chemotaxis system with any chemical diffusion, J. Differential Equations, 261 (2016), 2650-2669. doi: 10.1016/j.jde.2016.05.008. Google Scholar
[31]	A.I. Muñoz and J. I. Tello, Mathematical analysis and numerical simulation of a model of morphogenesis, Mathematical Biosciences and Engineering, 8 (2011), 1035-1059. doi: 10.3934/mbe.2011.8.1035. Google Scholar
[32]	M. Negreanu and J. I. Tello, On a two species chemotaxis model with slow chemical diffusion, SIAM J. Math. Anal., 46 (2014), 3761-3781. doi: 10.1137/140971853. Google Scholar
[33]	M. Negreanu and J. I. Tello, On a comparison method to reaction-diffusion systems and its applications to chemotaxis, Discrete and Continuous Dynamical Systems-Series B, 18 (2013), 2669-2688. doi: 10.3934/dcdsb.2013.18.2669. Google Scholar
[34]	M. Negreanu and J. I. Tello, On a two species chemotaxis model with slow chemical diffusion, SIAM J. Math. Anal., 46 (2014), 3761-3781. doi: 10.1137/140971853. Google Scholar
[35]	M. Negreanu and J. I. Tello, Asymptotic stability of a two species chemotaxis system with non-diffusive chemoattractant, J. Differential Equations, 258 (2015), 1592-1617. doi: 10.1016/j.jde.2014.11.009. Google Scholar
[36]	C. S. Patlak, Random walk with persistence and external bias, The Bulletin of Mathematical Biophysics, 15 (1953), 311-338. doi: 10.1007/BF02476407. Google Scholar
[37]	H. G. Othmer and A. Stevens, Aggregation, blowup, and collapse: The ABC$^{\prime}$s of taxis in reinforced random walks, SIAM J. Appl. Math., 57 (1997), 1044-1081. Google Scholar
[38]	B. D. Sleeman and H. A. Levine, Partial differential equations of chemotaxis and angiogenesis, Math. Methods Appl. Sci., 24 (2001), 405-426. doi: 10.1002/mma.212. Google Scholar
[39]	B. D. Sleeman and H. A. Levine, Partial differential equations of chemotaxis and angiogenesis, SIAM J. Appl. Math., 65 (2005), 790-817. doi: 10.1137/S0036139902415117. Google Scholar
[40]	A. Stevens, The derivation of chemotaxis equations as limit dynamics of moderately interacting stochactic many-particle systems, SIAM J. Appl. Math., 61 (2000), 183-212. doi: 10.1137/S0036139998342065. Google Scholar
[41]	C. Stinner, J. I. Tello and M. Winkler, Mathematical analysis of a model of chemotaxis arising from morphogenesis, Math. Methods Appl. Sci., 35 (2012), 445-465. doi: 10.1002/mma.1573. Google Scholar
[42]	C. Stinner, J. I. Tello and M. Winkler, Competitive exclusion in a two-species chemotaxis model, Journal of Mathematical Biology, 68 (2014), 1607-1626. doi: 10.1007/s00285-013-0681-7. Google Scholar
[43]	T. Suzuki, Mathematical models of tumor growth systems, Mathematica Bohemica, 137 (2012), 201-218. Google Scholar
[44]	T. Suzuki and R. Takahashi, Global in time solution to a class of tumour growth systems, Adv. Math. Sci. Appl., 19 (2009), 503-524. Google Scholar
[45]	J. I. Tello, Mathematical analysis of a model of Morphogenesis, Discrete and Continuous Dynamical Systems - Serie A., 25 (2009), 343-361. doi: 10.3934/dcds.2009.25.343. Google Scholar
[46]	J. I. Tello and M. Winkler, Stabilization in a two-species chemotaxis system with a logistic source, Nonlinearity, 25 (2012), 1413-1425. doi: 10.1088/0951-7715/25/5/1413. Google Scholar
[47]	A. M. Turing, The chemical basis of morphogenesis, Philosophical Transactions of the Royal Society of London, Series B, 237 (1952), 37-72. doi: 10.1098/rstb.1952.0012. Google Scholar
[48]	X. Wang and Y. Wu, Qualitative analysis on a chemotactic diffusion model for two species competing for a limited resource, Quart. Appl. Math., 60 (2002), 505-531. doi: 10.1090/qam/1914439. Google Scholar

show all references

References:

[1]	A. R. A. Anderson and M. A. I. Chaplain, Continuous and discrete mathematical models of tumor-induced angiogenesis, Bull. Math. Biology, 60 (1998), 857-899. doi: 10.1006/bulm.1998.0042. Google Scholar
[2]	T. Black, Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals, Discrete& Continuous Dynamical Systems-Series B, 22 (2017), 1253-1272. doi: 10.3934/dcdsb.2017061. Google Scholar
[3]	T. Black, J. Lankeit and M. Mizukami, On the weakly competitive case in a two-species chemotaxis model, IMA J Appl Math., 81 (2016), 860-876. doi: 10.1093/imamat/hxw036. Google Scholar
[4]	T. Bollenbach, K. Kruse, P. Pantazis, M. González-Gaitán and F. Jülicher, Morphogen transport in Epithelia, Physical Review E Stat Nonlin Soft Matter Phys, 75 (2007), 011901. doi: 10.1103/PhysRevE.75.011901. Google Scholar
[5]	N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Mathematical Models and Methods in Applied Sciences, 25 (2015), 1663-1763. doi: 10.1142/S021820251550044X. Google Scholar
[6]	C. Conca and E. Espejo, Threshold condition for global existence and blow-up to a radially symmetric drift-diffusion system, Appl. Math. Lett., 25 (2012), 352-356. doi: 10.1016/j.aml.2011.09.013. Google Scholar
[7]	L. Edelstein-Keshet, Mathematical Models in Biology, Society for Industrial and Applied Mathematics, Philadelphia 2005. doi: 10.1137/1.9780898719147. Google Scholar
[8]	A. Fasano, A. Mancini and M. Primicerio, Equilibrium of two populations subject to chemotaxis, Math. Models Methods Appl. Sci., 14 (2004), 503-533. doi: 10.1142/S0218202504003337. Google Scholar
[9]	M. A. Fontelos, A. Friedman and B. Hu, Mathematical analysis of a model for the initiation of angiogenesis, SIAM Math. Anal., 33 (2002), 1330-1355. doi: 10.1137/S0036141001385046. Google Scholar
[10]	A. Friedman and J. I. Tello, Stability of solutions of chemotaxis equations in reinforced random walks, J. Math. Anal. Appl., 272 (2002), 138-163. doi: 10.1016/S0022-247X(02)00147-6. Google Scholar
[11]	M. Hirata, S. Kurima, M. Mizukami and T. Yokota, Boundedness and stabilization in a twodimensional two-species chemotaxis-Navier- Stokes system with competitive kinetics, Journal of Differential Equations, 263 (2017), 470-490. doi: 10.1016/j.jde.2017.02.045. Google Scholar
[12]	D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, Jahresber. Dtsch. Math. Ver., 105 (2003), 103-165. Google Scholar
[13]	D. Horstmann, Generalizing the Keller-Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, J. Nonlinear Sci., 21 (2011), 231-270. doi: 10.1007/s00332-010-9082-x. Google Scholar
[14]	E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5. Google Scholar
[15]	E. F. Keller and L. A. Segel, A model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6. Google Scholar
[16]	H. Kielhöfer, Bifurcation Theory, Springer, 2004. doi: 10.1007/b97365. Google Scholar
[17]	P. Krzyżanowski, P. Laurençot and D. Wrzosek, Mathematical models of receptor-mediated transport of morphogens, M3AS, 20 (2010), 2021-2052. doi: 10.1142/S0218202510004866. Google Scholar
[18]	A. Kubo, H. Hoshino and K. Kimura, Global existence and asymptotic behaviour of solutions for nonlinear evolution equations related to a tumour invasion, Proceedings of 10th AIMS Conference, (2015), 733-744. doi: 10.3934/proc.2015.0733. Google Scholar
[19]	A. Kubo and T. Suzuki, Mathematical models of tumour angiogenesis, Journal of Computational and Applied Mathematics, 204 (2007), 48-55. doi: 10.1016/j.cam.2006.04.027. Google Scholar
[20]	A. Kubo and J. I. Tello, Mathematical analysis of a model of chemotaxis with competition terms, Differential and Integral Equations, 29 (2016), 441-454. Google Scholar
[21]	D. A. Lauffenburger, Quantitative studies of bacterial chemotaxis and microbial population dynamics, Microb. Ecol., 22 (1991), 175-185. doi: 10.1007/BF02540222. Google Scholar
[22]	H. A. Levine and B. D. Sleeman, A system of reaction diffusion equation arising in the theory of reinforced random walks, SIAM J. Appl. Math., 57 (1997), 683-730. doi: 10.1137/S0036139995291106. Google Scholar
[23]	H. A. Levine and J. Renclawowicz, Singularity formation in chemotaxis - a conjecture of nagai, SIAM J. Appl. Math., 65 (2004), 336-360. doi: 10.1137/S0036139903431725. Google Scholar
[24]	H. A. Levine, B. P. Sleeman and N. Nilsen-Hamilton, A mathematical modeling for the roles of pericytes and macrophages in the initiation of angiogenesis I. The role of protease inhibitors in preventing angiogenesis, Mathematical Biosciences, 168 (2000), 75-115. doi: 10.1016/S0025-5564(00)00034-1. Google Scholar
[25]	Y. Li, K. Lin and C. Mu, Boundedness and asymptotic behavior of solutions to a chemotaxishaptotaxis model in high dimensions, Applied Mathematics Letters, 50 (2015), 91-97. doi: 10.1016/j.aml.2015.06.010. Google Scholar
[26]	M. Malogrosz, Well-posedness and asymptotic behavior of a multidimensonal model of morphogen transport, J. Evol. Eq., 12 (2012), 353-366. doi: 10.1007/s00028-012-0135-5. Google Scholar
[27]	M. Małogrosz, A model of morphogen transport in the presence of glypicans II, Journal of Mathematical Analysis and Applications, 433 (2016), 642-680. doi: 10.1016/j.jmaa.2015.07.053. Google Scholar
[28]	J. H. Merking, D. J. Needham and B. D. Sleeman, A mathematical model for the spread of morphogens with density dependent chemosensitivity, Nonlinearity, 18 (2005), 2745-2773. doi: 10.1088/0951-7715/18/6/018. Google Scholar
[29]	J. H. Merking and B. D. Sleeman, On the spread of morphogens, J. Math. Biol., 51 (2005), 1-17. doi: 10.1007/s00285-004-0308-0. Google Scholar
[30]	M. Mizukami and T. Yokota, Global existence and asymptotic stability of solutions to a twospecies chemotaxis system with any chemical diffusion, J. Differential Equations, 261 (2016), 2650-2669. doi: 10.1016/j.jde.2016.05.008. Google Scholar
[31]	A.I. Muñoz and J. I. Tello, Mathematical analysis and numerical simulation of a model of morphogenesis, Mathematical Biosciences and Engineering, 8 (2011), 1035-1059. doi: 10.3934/mbe.2011.8.1035. Google Scholar
[32]	M. Negreanu and J. I. Tello, On a two species chemotaxis model with slow chemical diffusion, SIAM J. Math. Anal., 46 (2014), 3761-3781. doi: 10.1137/140971853. Google Scholar
[33]	M. Negreanu and J. I. Tello, On a comparison method to reaction-diffusion systems and its applications to chemotaxis, Discrete and Continuous Dynamical Systems-Series B, 18 (2013), 2669-2688. doi: 10.3934/dcdsb.2013.18.2669. Google Scholar
[34]	M. Negreanu and J. I. Tello, On a two species chemotaxis model with slow chemical diffusion, SIAM J. Math. Anal., 46 (2014), 3761-3781. doi: 10.1137/140971853. Google Scholar
[35]	M. Negreanu and J. I. Tello, Asymptotic stability of a two species chemotaxis system with non-diffusive chemoattractant, J. Differential Equations, 258 (2015), 1592-1617. doi: 10.1016/j.jde.2014.11.009. Google Scholar
[36]	C. S. Patlak, Random walk with persistence and external bias, The Bulletin of Mathematical Biophysics, 15 (1953), 311-338. doi: 10.1007/BF02476407. Google Scholar
[37]	H. G. Othmer and A. Stevens, Aggregation, blowup, and collapse: The ABC$^{\prime}$s of taxis in reinforced random walks, SIAM J. Appl. Math., 57 (1997), 1044-1081. Google Scholar
[38]	B. D. Sleeman and H. A. Levine, Partial differential equations of chemotaxis and angiogenesis, Math. Methods Appl. Sci., 24 (2001), 405-426. doi: 10.1002/mma.212. Google Scholar
[39]	B. D. Sleeman and H. A. Levine, Partial differential equations of chemotaxis and angiogenesis, SIAM J. Appl. Math., 65 (2005), 790-817. doi: 10.1137/S0036139902415117. Google Scholar
[40]	A. Stevens, The derivation of chemotaxis equations as limit dynamics of moderately interacting stochactic many-particle systems, SIAM J. Appl. Math., 61 (2000), 183-212. doi: 10.1137/S0036139998342065. Google Scholar
[41]	C. Stinner, J. I. Tello and M. Winkler, Mathematical analysis of a model of chemotaxis arising from morphogenesis, Math. Methods Appl. Sci., 35 (2012), 445-465. doi: 10.1002/mma.1573. Google Scholar
[42]	C. Stinner, J. I. Tello and M. Winkler, Competitive exclusion in a two-species chemotaxis model, Journal of Mathematical Biology, 68 (2014), 1607-1626. doi: 10.1007/s00285-013-0681-7. Google Scholar
[43]	T. Suzuki, Mathematical models of tumor growth systems, Mathematica Bohemica, 137 (2012), 201-218. Google Scholar
[44]	T. Suzuki and R. Takahashi, Global in time solution to a class of tumour growth systems, Adv. Math. Sci. Appl., 19 (2009), 503-524. Google Scholar
[45]	J. I. Tello, Mathematical analysis of a model of Morphogenesis, Discrete and Continuous Dynamical Systems - Serie A., 25 (2009), 343-361. doi: 10.3934/dcds.2009.25.343. Google Scholar
[46]	J. I. Tello and M. Winkler, Stabilization in a two-species chemotaxis system with a logistic source, Nonlinearity, 25 (2012), 1413-1425. doi: 10.1088/0951-7715/25/5/1413. Google Scholar
[47]	A. M. Turing, The chemical basis of morphogenesis, Philosophical Transactions of the Royal Society of London, Series B, 237 (1952), 37-72. doi: 10.1098/rstb.1952.0012. Google Scholar
[48]	X. Wang and Y. Wu, Qualitative analysis on a chemotactic diffusion model for two species competing for a limited resource, Quart. Appl. Math., 60 (2002), 505-531. doi: 10.1090/qam/1914439. Google Scholar

[1]	Ling Liu, Jiashan Zheng. Global existence and boundedness of solution of a parabolic-parabolic-ODE chemotaxis-haptotaxis model with (generalized) logistic source. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3357-3377. doi: 10.3934/dcdsb.2018324
[2]	Karl Kunisch, Sérgio S. Rodrigues. Oblique projection based stabilizing feedback for nonautonomous coupled parabolic-ode systems. Discrete & Continuous Dynamical Systems, 2019, 39 (11) : 6355-6389. doi: 10.3934/dcds.2019276
[3]	Tomasz Cieślak, Kentarou Fujie. Global existence in the 1D quasilinear parabolic-elliptic chemotaxis system with critical nonlinearity. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 165-176. doi: 10.3934/dcdss.2020009
[4]	T. Hillen, K. Painter, Christian Schmeiser. Global existence for chemotaxis with finite sampling radius. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 125-144. doi: 10.3934/dcdsb.2007.7.125
[5]	Mengyao Ding, Wei Wang. Global boundedness in a quasilinear fully parabolic chemotaxis system with indirect signal production. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4665-4684. doi: 10.3934/dcdsb.2018328
[6]	Wei Wang, Yan Li, Hao Yu. Global boundedness in higher dimensions for a fully parabolic chemotaxis system with singular sensitivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3663-3669. doi: 10.3934/dcdsb.2017147
[7]	Ke Lin, Chunlai Mu. Global dynamics in a fully parabolic chemotaxis system with logistic source. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 5025-5046. doi: 10.3934/dcds.2016018
[8]	Daniela Giachetti, Maria Michaela Porzio. Global existence for nonlinear parabolic equations with a damping term. Communications on Pure & Applied Analysis, 2009, 8 (3) : 923-953. doi: 10.3934/cpaa.2009.8.923
[9]	Sainan Wu, Junping Shi, Boying Wu. Global existence of solutions to an attraction-repulsion chemotaxis model with growth. Communications on Pure & Applied Analysis, 2017, 16 (3) : 1037-1058. doi: 10.3934/cpaa.2017050
[10]	Huanhuan Qiu, Shangjiang Guo. Global existence and stability in a two-species chemotaxis system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1569-1587. doi: 10.3934/dcdsb.2018220
[11]	Radek Erban, Hyung Ju Hwang. Global existence results for complex hyperbolic models of bacterial chemotaxis. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1239-1260. doi: 10.3934/dcdsb.2006.6.1239
[12]	Abelardo Duarte-Rodríguez, Lucas C. F. Ferreira, Élder J. Villamizar-Roa. Global existence for an attraction-repulsion chemotaxis fluid model with logistic source. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 423-447. doi: 10.3934/dcdsb.2018180
[13]	Johannes Lankeit, Yulan Wang. Global existence, boundedness and stabilization in a high-dimensional chemotaxis system with consumption. Discrete & Continuous Dynamical Systems, 2017, 37 (12) : 6099-6121. doi: 10.3934/dcds.2017262
[14]	Guoqiang Ren, Heping Ma. Global existence in a chemotaxis system with singular sensitivity and signal production. Discrete & Continuous Dynamical Systems - B, 2022, 27 (1) : 343-360. doi: 10.3934/dcdsb.2021045
[15]	Etsushi Nakaguchi, Koichi Osaki. Global solutions and exponential attractors of a parabolic-parabolic system for chemotaxis with subquadratic degradation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2627-2646. doi: 10.3934/dcdsb.2013.18.2627
[16]	Rachidi B. Salako, Wenxian Shen. Existence of traveling wave solutions to parabolic-elliptic-elliptic chemotaxis systems with logistic source. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 293-319. doi: 10.3934/dcdss.2020017
[17]	Francesca R. Guarguaglini. Global solutions for a chemotaxis hyperbolic-parabolic system on networks with nonhomogeneous boundary conditions. Communications on Pure & Applied Analysis, 2020, 19 (2) : 1057-1087. doi: 10.3934/cpaa.2020049
[18]	Hong Yi, Chunlai Mu, Shuyan Qiu, Lu Xu. Global boundedness of radial solutions to a parabolic-elliptic chemotaxis system with flux limitation and nonlinear signal production. Communications on Pure & Applied Analysis, 2021, 20 (11) : 3825-3849. doi: 10.3934/cpaa.2021133
[19]	Zhengce Zhang, Yan Li. Global existence and gradient blowup of solutions for a semilinear parabolic equation with exponential source. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 3019-3029. doi: 10.3934/dcdsb.2014.19.3019
[20]	Feng Li, Yuxiang Li. Global existence of weak solution in a chemotaxis-fluid system with nonlinear diffusion and rotational flux. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5409-5436. doi: 10.3934/dcdsb.2019064

American Institute of Mathematical Sciences

On a Parabolic-ODE system of chemotaxis

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

On a Parabolic-ODE system of chemotaxis

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors