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# Global boundedness of solutions to the two-dimensional forager-exploiter model with logistic source

• * Corresponding author: Qiao Xin
The second author is supported by NSFC [No. 11771062, 11971082], Fundamental Research Funds for the Central Universities [No. 2019CDJCYJ001], and Chongqing Key Laboratory of Analytic Mathematics and Applications. The third author is supported by the Youth Doctor Science and Technology Talent Training Project of Xinjiang Uygur Autonomous Region [No. 2017Q087]
• This paper deals with the global boundedness of solutions to the forager-exploiter model with logistic sources

$\begin{equation*} \left\{ \begin{array}{lll} u_t = \Delta u- \nabla\cdot(u\nabla w) + \mu_1 (u-u^m), &x \in \Omega, t>0,\\ v_t = \Delta v - \nabla\cdot(v\nabla u) + \mu_2 ( v-v^l), &x\in \Omega, t>0,\\ w_t = \Delta w - \lambda(u+v)w - \mu w + r(x,t), & x\in \Omega, t>0, \end{array} \right. \end{equation*}$

under homogeneous Neumann boundary conditions in a smoothly bounded domain $\Omega \subset R^2$, where the constants $\mu$, $\mu_1$, $\mu_2$, $\lambda$, $m$ and $l$ are positive. We prove that the corresponding initial-boundary value problem possesses a global classical solution that is uniformly bounded under conditions $2\leq m < 3$, $l \geq 3$, $r(x,t) \in C^1(\overline{\Omega}\times[0,\infty))\cup L^{\infty}(\Omega\times(0,\infty))$ and the smooth nonnegative initial functions, which improves the results obtained by Wang and Wang (MMMAS 2020).

Mathematics Subject Classification: 35A01, 35K55.

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