Boundedness in a quasilinear attraction–repulsion chemotaxis system with variable logistic source

This paper deals with a quasilinear attraction–repulsion chemotaxis system with a source term of variable logistic type $ u_t = \nabla\cdot(\phi(u)\nabla u)-\nabla\cdot(\psi(u)\nabla v)+\nabla\cdot(\varphi(u)\nabla w)+g(u) $, $ \tau_1v_t = \Delta v-v+u $, $ -\Delta w = -w+u $ in a smooth bounded domain $ \Omega\subset \mathbb R^n $ ($ n\ge1 $), and endowed with nonnegative initial data and homogeneous Neumann boundary conditions. Moreover, the logistic source verifies $ g(x, s)\leq\eta s^{k(x)}-\mu s^{m(x)} $, $ s > 0 $ with $ g(x, 0)\ge0 $, $ x\in\Omega $, where $ \eta\ge0 $, $ \mu > 0 $ are constants, $ k, m $ are measurable functions fulfilling $ 0\leq k^-: = \underset{x\in\Omega}{ess\ \inf} k(x)\leq k(x)\leq k^+: = \underset{x\in\Omega}{ess\ \sup} k(x) < +\infty $ and $ 1 < m^-: = \underset{x\in\Omega}{ess\ \inf} m(x)\leq m(x)\leq m^+: = \underset{x\in\Omega}{ess\ \sup} m(x) < +\infty $, as well as $ \phi, \psi $, and $ \varphi $ are regular functions satisfying $ c_1s^p\leq\phi(s) $, $ \psi(s)\leq c_2s^q $, and $ \underline c_3s^l\leq\varphi(s)\leq c_3s^l $ with $ p, q, l\in\mathbb R $, $ c_1, c_2, \underline c_3, c_3 > 0 $ and $ s\ge s_0 > 1 $. We show that when $ q = m^{−} − 1 $ and $ l\leq m^{−} − 1 $, there exists $ \mu^* > 0 $ such that if $ \mu > \mu^* $, then the corresponding initial-boundary value problem possesses a unique globally bounded classical solution. Moreover, the same conclusion holds true provided that $ q < m^{−} − 1 $ and $ l\leq m^{−} − 1 $ for any $ \mu > 0 $.