Global boundedness in an attraction-repulsion chemotaxis system with nonlinear productions and logistic source

Abstract

Abstract This paper deals with the quasilinear(\(\tau =0\)) and fully parabolic(\(\tau =1\)) attraction-repulsion chemotaxis system with nonlinear productions and logistic source, \(u_t = \newnabla{D(u)}{u} - \newnabla{\Phi (u)}{v} + \newnabla{\Psi (u)}{w} + f(u), v_t = \Delta v+\alpha {{u}^{k}}-\beta v, \tau w_t = \Delta w+\gamma {{u}^{l}}-\delta w, \tau \in \{0,1\},\) in bounded domain \(\Omega \subset {{\mathbb{R}}^{n}} \text{ } \newbrac{n \ge 1},\) subject to the homogeneous Neumann boundary conditions and initial conditions, \(D,\Phi ,\Psi \in {{C}^{2}}[0,\infty )\) nonnegative with \(D(s)\ge {{(s+1)}^{p}}\text{ for }s\ge 0,\) \(\Phi (s)\le \chi {{s}^{q}},\) \(\xi {{s}^{g}}\le \Psi (s),\text{ }s\ge {{s}_{0}}\) for \({{s}_{0}}>1.\) And the logistic source satisfying\(f(s)\le s(a-b{{s}^{d}}), \text{ } s>0, \text{ } f(0)\ge 0,\) and the nonlinear productions for the attraction and repulsion chemicals are described via \(\alpha {{u}^{k}} \text{ and } \gamma {{u}^{l}}\) respectively. When \(k=l=1\) , it is known that above system possesses a globally bounded solution in some cases. However, there has been no work in the case that \(k,l>0\). This paper develops global boundedness of the solution to the above system in some cases. And extends the global boundedness criteria established by Tian-He-Zheng(2016) for the quasilinear attraction-repulsion chemotaxis system.