December  2021, 41(12): 6001-6022. doi: 10.3934/dcds.2021104

Global wellposedness of nutrient-taxis systems derived by a food metric

1. 

Department of Mathematics, Dongguk University, Seoul, 04620, Republic of Korea

2. 

Department of Applied Mathematics and the Institute of Natural Sciences, Kyung Hee University, Yongin, 17104, Republic of Korea

3. 

National Institute for Mathematical Sciences, Daejeon, 34047, Republic of Korea

* Corresponding author: Sun-Ho Choi

Received  October 2020 Published  December 2021 Early access  June 2021

Fund Project: Jaewook Ahn is supported by National Research Foundation (NRF) of Korea (Grant no. 2018R1D1A1B07047465). Sun-Ho Choi is supported by National Research Foundation (NRF) of Korea (Grant no. 2017R1E1A1A03070692). Minha Yoo is supported by the National Institute for Mathematical Sciences (NIMS) granted by the Korean government (Grant no. NIMS-B21900000)

This paper deals with the nutrient-taxis system derived by a food metric. The system was proposed in [Sun-Ho Choi and Yong-Jung Kim: Chemotactic traveling waves by metric of food, SIAM J. Appl. Math. 75 (2015), 2268–2289] using geometric ideas without gradient sensing, and has a simple form but contains a singular diffusive coefficient on the equation for the organism side. To overcome the difficulty arising from this singular structure, we use a weighted $ L^{p} $-estimate involving a weighted Gagliardo-Nirenberg type inequality. In the one dimensional setting, it turns out that the system is shown to be globally well-posed in certain Sobolev spaces and the solutions are uniformly bounded. Moreover, the zero viscosity limit of the equation for the nutrient side is considered. For the same initial data and any given finite time interval, a diffusive solution converges to a non-diffusive solution when the diffusion coefficient vanishes.

Citation: Jaewook Ahn, Sun-Ho Choi, Minha Yoo. Global wellposedness of nutrient-taxis systems derived by a food metric. Discrete & Continuous Dynamical Systems, 2021, 41 (12) : 6001-6022. doi: 10.3934/dcds.2021104
References:
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J. Adler and M. Dahl, A method for measuring the motility of bacteria and for comparing random and non-random motility, Microbiology, 46 (1967), 161-173.   Google Scholar

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S.-H. Choi and Y.-J. Kim, A discrete velocity kinetic model with food metric: Chemotaxis traveling waves, Bull. Math. Biol., 79 (2017), 277-302.  doi: 10.1007/s11538-016-0235-4.  Google Scholar

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S.-H. Choi and Y.-J. Kim, Chemotactic traveling waves with compact support, J. Math. Anal. Appl., 488 (2020), 124090, 21 pp. doi: 10.1016/j.jmaa.2020.124090.  Google Scholar

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J. Fan and K. Zhao, Blow up criteria for a hyperbolic-parabolic system arising from chemotaxis, J. Math. Anal. Appl., 394 (2012), 687-695.  doi: 10.1016/j.jmaa.2012.05.036.  Google Scholar

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[24]

G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302.  Google Scholar

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F. Poupaud and J. Soler, Parabolic limit and stability of the Vlasov-Fokker-Planck system, Math. Models Methods Appl. Sci., 10 (2000), 1027-1045.  doi: 10.1142/S0218202500000525.  Google Scholar

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show all references

References:
[1]

J. Adler, Chemotaxis in bacteria, Science, 153 (1966), 708-716.   Google Scholar

[2]

J. Adler and M. Dahl, A method for measuring the motility of bacteria and for comparing random and non-random motility, Microbiology, 46 (1967), 161-173.   Google Scholar

[3]

N. D. Alikakos, $L^{p}$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868.  doi: 10.1080/03605307908820113.  Google Scholar

[4]

W. Alt, Orientation of cells migrating in a chemotactic gradient, In: Biological Growth and Spread, Springer, Berlin, Heidelberg, 38 (1980), 353–366.  Google Scholar

[5]

W. E. Boyce and R. C. DiPrima, Elementary Differential Equations and Boundary Value Problems, John Wiley & Sons Inc., New York, 1965.  Google Scholar

[6]

F. A. C. C. ChalubP. A. MarkowichB. Perthame and C. Schmeiser, Kinetic models for chemotaxis and their drift-diffusion limits, Monatsh. Math., 142 (2004), 123-141.  doi: 10.1007/s00605-004-0234-7.  Google Scholar

[7]

S.-H. Choi and Y.-J. Kim, Chemotactic traveling waves by metric of food, SIAM J. Appl. Math., 75 (2015), 2268-2289.  doi: 10.1137/15100429X.  Google Scholar

[8]

S.-H. Choi and Y.-J. Kim, A discrete velocity kinetic model with food metric: Chemotaxis traveling waves, Bull. Math. Biol., 79 (2017), 277-302.  doi: 10.1007/s11538-016-0235-4.  Google Scholar

[9]

S.-H. Choi and Y.-J. Kim, Chemotactic traveling waves with compact support, J. Math. Anal. Appl., 488 (2020), 124090, 21 pp. doi: 10.1016/j.jmaa.2020.124090.  Google Scholar

[10]

J. Fan and K. Zhao, Blow up criteria for a hyperbolic-parabolic system arising from chemotaxis, J. Math. Anal. Appl., 394 (2012), 687-695.  doi: 10.1016/j.jmaa.2012.05.036.  Google Scholar

[11]

M. A. FontelosA. Friedman and B. Hu, Mathematical analysis of a model for the initiation of angiogenesis, SIAM J. Math. Anal., 33 (2002), 1330-1355.  doi: 10.1137/S0036141001385046.  Google Scholar

[12]

J. GuoJ. XiaoH. Zhao and C. Zhu, Global solutions to a hyperbolic-parabolic coupled system with large initial data, Acta Math. Sci. Ser. B (Engl. Ed.), 29 (2009), 629-641.  doi: 10.1016/S0252-9602(09)60059-X.  Google Scholar

[13]

C. Hao, Global well-posedness for a multidimensional chemotaxis model in critical Besov spaces, Z. Angew. Math. Phys., 63 (2012), 825-834.  doi: 10.1007/s00033-012-0193-0.  Google Scholar

[14]

D. HilhorstY.-J. KimD. Kwon and T. N. Nguyen, Dispersal toward food: A study of a singular limit of an Allen-Cahn equation, J. Math. Biol., 76 (2018), 531-565.  doi: 10.1007/s00285-017-1150-5.  Google Scholar

[15]

E. F. Keller and L. A. Segel, Model for chemotaxis, Journal of Theoretical Biology, 30 (1971), 225-234.   Google Scholar

[16]

E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, Journal of Theoretical Biology, 30 (1971), 235-248.   Google Scholar

[17]

N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, Graduate Studies in Mathematics, 96. American Mathematical Society, Providence, RI, 2008. doi: 10.1090/gsm/096.  Google Scholar

[18]

O. A. Ladyžhenskaya, V. A. Solonnikov and N. N. Ural\'ceva, Linear and Quasi-Linear Equations of Parabolic Type, 23, (Providence, RI: American Mathematical Society), 1968.  Google Scholar

[19]

H. A. Levine and B. D. Sleeman, A system of reaction diffusion equations arising in the theory of reinforced random walks, SIAM J. Appl. Math., 57 (1997), 683-730.  doi: 10.1137/S0036139995291106.  Google Scholar

[20]

D. LiT. Li and K. Zhao, On a hyperbolic-parabolic system modeling chemotaxis, Math. Models Methods Appl. Sci., 21 (2011), 1631-1650.  doi: 10.1142/S0218202511005519.  Google Scholar

[21]

D. LiR. Pan and K. Zhao, Quantitative decay of a hybrid type chemotaxis model with large data, Nonlinearity, 28 (2015), 2181-2210.  doi: 10.1088/0951-7715/28/7/2181.  Google Scholar

[22]

T. LiR. Pan and K. Zhao, Global dynamics of a hyperbolic-parabolic model arising from chemotaxis, SIAM J. Appl. Math., 72 (2012), 417-443.  doi: 10.1137/110829453.  Google Scholar

[23]

H. Li and K. Zhao, Initial boundary value problems for a system of hyperbolic balance laws arising from chemotaxis, J. Differential Equations, 258 (2015), 302-338.  doi: 10.1016/j.jde.2014.09.014.  Google Scholar

[24]

G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302.  Google Scholar

[25]

F. Poupaud and J. Soler, Parabolic limit and stability of the Vlasov-Fokker-Planck system, Math. Models Methods Appl. Sci., 10 (2000), 1027-1045.  doi: 10.1142/S0218202500000525.  Google Scholar

[26]

Y. TaoL. Wang and Z.-A. Wang, Large-time behavior of a parabolic-parabolic chemotaxis model with logarithmic sensitivity in one dimension, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 821-845.  doi: 10.3934/dcdsb.2013.18.821.  Google Scholar

[27]

Z.-A. WangZ. Xiang and P. Yu, Asymptotic dynamics on a singular chemotaxis system modeling onset of tumor angiogenesis, J. Differential Equations, 260 (2016), 2225-2258.  doi: 10.1016/j.jde.2015.09.063.  Google Scholar

[28]

Y. ZhangZ. Tan and M.-B. Sun, Global existence and asymptotic behavior of smooth solutions to a coupled hyperbolic-parabolic system, Nonlinear Anal. Real World Appl., 14 (2013), 465-482.  doi: 10.1016/j.nonrwa.2012.07.009.  Google Scholar

[29]

M. Zhang and C. Zhu, Global existence of solutions to a hyperbolic-parabolic system, Proc. Amer. Math. Soc., 135 (2007), 1017-1027.  doi: 10.1090/S0002-9939-06-08773-9.  Google Scholar

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