Mathematical Biosciences and Engineering

2023, Issue 5: 7922-7942. doi: 10.3934/mbe.2023343
Theory article

Global boundedness of a higher-dimensional chemotaxis system on alopecia areata

• Received: 01 December 2022 Revised: 13 January 2023 Accepted: 07 February 2023 Published: 23 February 2023
• This paper mainly focuses on the dynamics behavior of a three-component chemotaxis system on alopecia areata

$\begin{equation*} \left\{ \begin{array}{lll} u_t = \Delta{u}-\chi_1\nabla\cdot(u\nabla{w})+w-\mu_1u^2, &x\in\Omega, t>0, \\ v_t = \Delta{v}-\chi_2\nabla\cdot(v\nabla{w})+w+ruv-\mu_2v^2, &x\in \Omega, t>0, \\ w_t = \Delta{w}+u+v-w, &x\in \Omega, t>0, \\ \frac{\partial{u}}{\partial{\nu}} = \frac{\partial{v}}{\partial{\nu}} = \frac{\partial{w}}{\partial{\nu}} = 0, &x\in \partial \Omega, t>0, \\ u(x, 0) = u_0(x), \ v(x, 0) = v_0(x), \ w(x, 0) = w_0(x), &x\in \Omega, \ \end{array} \right. \end{equation*}$

where $\Omega\subset\mathbb{R}^n$ $(n \geq 4)$ is a bounded convex domain with smooth boundary $\partial\Omega$, the parameters $\chi_i$, $\mu_i$ $(i = 1, 2)$, and $r$ are positive. We show that this system exists a globally bounded classical solution if $\mu_i\; (i = 1, 2)$ is large enough. This result extends the corresponding results which were obtained by Lou and Tao (JDE, 2021) to the higher-dimensional case.

Citation: Wenjie Zhang, Lu Xu, Qiao Xin. Global boundedness of a higher-dimensional chemotaxis system on alopecia areata[J]. Mathematical Biosciences and Engineering, 2023, 20(5): 7922-7942. doi: 10.3934/mbe.2023343

Related Papers:

• This paper mainly focuses on the dynamics behavior of a three-component chemotaxis system on alopecia areata

$\begin{equation*} \left\{ \begin{array}{lll} u_t = \Delta{u}-\chi_1\nabla\cdot(u\nabla{w})+w-\mu_1u^2, &x\in\Omega, t>0, \\ v_t = \Delta{v}-\chi_2\nabla\cdot(v\nabla{w})+w+ruv-\mu_2v^2, &x\in \Omega, t>0, \\ w_t = \Delta{w}+u+v-w, &x\in \Omega, t>0, \\ \frac{\partial{u}}{\partial{\nu}} = \frac{\partial{v}}{\partial{\nu}} = \frac{\partial{w}}{\partial{\nu}} = 0, &x\in \partial \Omega, t>0, \\ u(x, 0) = u_0(x), \ v(x, 0) = v_0(x), \ w(x, 0) = w_0(x), &x\in \Omega, \ \end{array} \right. \end{equation*}$

where $\Omega\subset\mathbb{R}^n$ $(n \geq 4)$ is a bounded convex domain with smooth boundary $\partial\Omega$, the parameters $\chi_i$, $\mu_i$ $(i = 1, 2)$, and $r$ are positive. We show that this system exists a globally bounded classical solution if $\mu_i\; (i = 1, 2)$ is large enough. This result extends the corresponding results which were obtained by Lou and Tao (JDE, 2021) to the higher-dimensional case.

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