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

Lu Xu; Chunlai Mu; Qiao Xin

doi:10.3934/dcds.2020396

Article Contents

2021, Volume 41, Issue 7: 3031-3043. Doi: 10.3934/dcds.2020396

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

1.
College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China
2.
College of Mathematics and Statistics, Yili Normal University, Yining 835000, China

^* Corresponding author: Qiao Xin
^* Corresponding author: Qiao Xin

Received: May 31, 2020

Revised: September 30, 2020

Early access: December 2020

Published: July 2021

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]

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Abstract

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).

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Mathematics Subject Classification: 35A01, 35K55.

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