Electron. J. Differential Equations, Vol. 2025 (2025), No. 01, pp. 1-17.
Existence and boundedness of solutions for a parabolic-parabolic
predator-prey model
Fengxiang Zhao, Haotian Tang, Jiashan Zheng, Kaiqiang Li
Abstract:
This article concerns the fully parabolic pursuit-prey chemotaxis system
$$\displaylines{
u_t=\Delta u-\chi\nabla\cdot\left(u\nabla w\right)
+u\left(\lambda_1-\mu_1 u^{r_1-1}+av\right), \quad x\in\Omega,\; t>0,\cr
v_t=\Delta v+\xi\nabla\cdot\left(v\nabla z\right)+v\left(\lambda_2-\mu_2
v^{r_2-1}-bu\right),\quad x\in\Omega,\; t>0,\cr
w_t=\Delta w-w+v,\quad x\in\Omega,\; t>0,\cr
z_t=\Delta z-z+u, \quad x\in\Omega,\; t>0,
}$$
in a bounded domain \(\Omega\subset\mathbb{R}^{N}\) \((N\geq1)\) with
homogeneous Neumann boundary conditions, where \(\chi\), \(\xi\),
\(\lambda_i\), \(\mu_i\), \(a\), \(b\) are positive constants and
\(r_i>1\) \((i=1,2)\). We show that if \((r_1-1)(r_2-1)\geq1\),
the above system exists a unique global and bounded classical solution
for all appropriately regular nonnegative initial data, which extends
the previous global existence result in Qi and Ke [13].
Submitted April 11, 2024. Published January 4, 2025.
Math Subject Classifications: 35K20, 35K55, 92C17.
Key Words: Pursuit-evasion; parabolic-parabolic; boundedness; classical solution.
DOI: 10.58997/ejde.2025.01
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Fengxiang Zhao
School of Mathematical and Informational Sciences
Yantai University
Yantai, 264005, Shandong, China
email: zfx2037@163.com
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Haotian Tang
Department of Mathematics
Faculty of Science and Technology
University of Macau
Taipa, Macau, China
email: haotiantang2022@163.com
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Jiashan Zheng
School of Mathematical and Informational Sciences
Yantai University
Yantai, 264005, Shandong, China
email: zhengjiashan2008@163.com
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Kaiqiang Li
School of Mathematical and Informational Sciences
Yantai University
Yantai, 264005, Shandong, China
email: kaiqiangli19@163.com
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