Global solvability of a chemotaxis-haptotaxis model in the whole 2-d space

Meng Liu; Yuxiang Li; Meng Liu; Yuxiang Li

doi:10.3934/mbe.2023327

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 4: 7565-7593. doi: 10.3934/mbe.2023327

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Global solvability of a chemotaxis-haptotaxis model in the whole 2-d space

Meng Liu ,
Yuxiang Li ^,

School of Mathematics, Southeast University, Nanjing 211189, China

Academic Editor: Zhi-An Wang

Received: 27 October 2022 Revised: 07 February 2023 Accepted: 14 February 2023 Published: 17 February 2023

This paper investigates a two-dimensional chemotaxis-haptotaxis model

$ \begin{eqnarray*} \left\{\begin{array}{lll} u_t = \Delta u-\chi\nabla\cdot(u\nabla v)-\xi\nabla\cdot(u\nabla w),&{} x\in\mathbb{R}^2,\ t>0,\\ v_t = \Delta v-v+u,&{}x\in\mathbb{R}^2,\ t>0,\\ w_t = -vw,&{}x\in\mathbb{R}^2,\ t>0, \end{array}\right. \end{eqnarray*} $

where $ \chi $ and $ \xi $ are positive parameters. It is proved that, for any suitable smooth initial data $ (u_0, v_0, w_0) $, this model admits a unique global strong solution if $ \left\|u_0\right\|_{L^1} < \frac{8 \pi}{\chi} $. Compared to the result by Calvez and Corrias (Calvez and Corrias, 2008 ^[1]), we can see that the haptotaxis effect is almost negligible in terms of global existence, which is consistent with the result of bounded domain (Jin and Xiang, 2021 ^[2]). Moreover, to the best of our knowledge, this is the first analytical work for the well-posedness of chemotaxis-haptotaxis system in the whole space.
- chemotaxis,
- haptotaxis,
- global existence,
- whole space,
- Cauchy problem
Citation: Meng Liu, Yuxiang Li. Global solvability of a chemotaxis-haptotaxis model in the whole 2-d space[J]. Mathematical Biosciences and Engineering, 2023, 20(4): 7565-7593. doi: 10.3934/mbe.2023327

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Abstract

This paper investigates a two-dimensional chemotaxis-haptotaxis model

$ \begin{eqnarray*} \left\{\begin{array}{lll} u_t = \Delta u-\chi\nabla\cdot(u\nabla v)-\xi\nabla\cdot(u\nabla w),&{} x\in\mathbb{R}^2,\ t>0,\\ v_t = \Delta v-v+u,&{}x\in\mathbb{R}^2,\ t>0,\\ w_t = -vw,&{}x\in\mathbb{R}^2,\ t>0, \end{array}\right. \end{eqnarray*} $

where $ \chi $ and $ \xi $ are positive parameters. It is proved that, for any suitable smooth initial data $ (u_0, v_0, w_0) $, this model admits a unique global strong solution if $ \left\|u_0\right\|_{L^1} < \frac{8 \pi}{\chi} $. Compared to the result by Calvez and Corrias (Calvez and Corrias, 2008 ^[1]), we can see that the haptotaxis effect is almost negligible in terms of global existence, which is consistent with the result of bounded domain (Jin and Xiang, 2021 ^[2]). Moreover, to the best of our knowledge, this is the first analytical work for the well-posedness of chemotaxis-haptotaxis system in the whole space.

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