Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 01, pp. 1-17.

Title: Existence and boundedness of solutions for a parabolic-parabolic 
predator-prey model

Authors: Fengxiang Zhao (Yantai Univ., Yantai, Shandong, China)
         Haotian Tang (Univ. of Macau, Taipa, Macau, China)
         Jiashan Zheng (Yantai Univ., Yantai, Shandong, China)
         Kaiqiang Li (Yantai Univ., Yantai, Shandong, China)
  
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 04, 2025.
Math Subject Classifications: 35K20, 35K55, 92C17.
Key Words: Pursuit-evasion; parabolic-parabolic; boundedness; classical solution.