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

Jaewook Ahn; Sun-Ho Choi; Minha Yoo

doi:10.3934/dcds.2021104

Article Contents

2021, Volume 41, Issue 12: 6001-6022. Doi: 10.3934/dcds.2021104 open access

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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
^* Corresponding author: Sun-Ho Choi

Received: September 30, 2020

Early access: June 2021

Published: December 2021

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)

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Abstract

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.

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Mathematics Subject Classification: Primary: 35M11, 35Q92; Secondary: 92C17.

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