Global boundedness of solutions to the two-dimensional forager-exploiter model with logistic source
Lü Xu, Chunlai Mu, Qiao Xin
Abstract
<p style='text-indent:20px;'>This paper deals with the global boundedness of solutions to the forager-exploiter model with logistic sources <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \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*} $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>under homogeneous Neumann boundary conditions in a smoothly bounded domain <inline-formula><tex-math id="M1">\begin{document}$ \Omega \subset R^2 $\end{document}</tex-math></inline-formula>, where the constants <inline-formula><tex-math id="M2">\begin{document}$ \mu $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ \mu_1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ \mu_2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ m $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ l $\end{document}</tex-math></inline-formula> are positive. We prove that the corresponding initial-boundary value problem possesses a global classical solution that is uniformly bounded under conditions <inline-formula><tex-math id="M8">\begin{document}$ 2\leq m < 3 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M9">\begin{document}$ l \geq 3 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M10">\begin{document}$ r(x,t) \in C^1(\overline{\Omega}\times[0,\infty))\cup L^{\infty}(\Omega\times(0,\infty)) $\end{document}</tex-math></inline-formula> and the smooth nonnegative initial functions, which improves the results obtained by Wang and Wang (MMMAS 2020).