Global boundedness in a parabolic-parabolic-elliptic attraction-repulsion chemotaxis system with nonlinear productions

Abstract

This paper investigates the global boundedness of the semilinear attraction-repulsion chemotaxis system with nonlinear productions and logistic source: \begin{document}$ \begin{cases} u_{t} = \Delta u-\chi\nabla\cdot(u\nabla v)+\xi\nabla\cdot(u\nabla w)+u(a-bu^s), \ (x, t)\in \Omega\times (0, T), \\[6pt] v_t = \Delta v+\alpha u^k-\beta v, \ (x, t)\in \Omega\times (0, T), \\[6pt] 0 = \Delta w+\gamma u^l-\delta w, \ (x, t)\in\Omega\times (0, T), \end{cases} $\end{document} with $ \alpha, \beta, \gamma, \delta, \chi, \xi, a, b, s, k, l>0 $, subject to the homogeneous Neumann boundary conditions in a bounded domain $ \Omega\subset\mathbb{R}^n $ ($ n\ge1 $) with smooth boundary. Hong et al. (J. Math. Anal. Appl. 484: 12703, 2020) have established the global boundedness criteria for the parabolic-elliptic-elliptic attraction-repulsion chemotaxis system. With the help of the maximum Sobolev regularity, this paper generalizes the results to the parabolic-parabolic-elliptic case. It is proved that if one of the random diffusion, logistic source and repulsion mechanisms dominates the attraction with $ \max\{l, s, \frac{2}{n}\}>k $, the solutions would be globally bounded. Under the balance situations, the boundedness of solutions will be determined by the sizes of the coefficients involved, but Hong et al only considered the balance of logistic source and repulsion mechanisms with $ \max\{l, s\} = k $, the balance produced by random diffusion was not taken into account, this paper consider the balance of random diffusion, logistic source and repulsion mechanisms with $ \max\{l, s, \frac{2}{n}\} = k $, the clear characterization of the global boundedness criteria is obtained for the parabolic-parabolic-elliptic attraction-repulsion chemotaxis system.