Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source

This paper deals with the global existence and boundedness of the solutions for the chemotaxis system with logistic source \begin{eqnarray*} \left\{ \begin{array}{llll} u_t=\nabla\cdot(\phi(u)\nabla u)-\nabla\cdot(\varphi(u)\nabla v)+f(u),\quad &x\in \Omega,\quad t>0,\\ v_t=\Delta v-v+u,\quad &x\in\Omega,\quad t>0,\\ \end{array} \right. \end{eqnarray*} under homogeneous Neumann boundary conditions in a convex smooth bounded domain $\Omega\subset \mathbb{R}^n (n\geq2),$ with non-negative initial data $u_0\in C^0(\overline{\Omega})$ and $v_0\in W^{1,\theta}{(\Omega)}$ (with some $\theta>n$). The nonlinearities $\phi$ and $\varphi$ are assumed to generalize the prototypes \begin{eqnarray*} \phi(u)=(u+1)^{-\alpha},\,\,\,\,\,\, \varphi(u)=u(u+1)^{\beta-1} \end{eqnarray*} with $\alpha\in \mathbb{R}$ and $\beta\in \mathbb{R}$. $f(u)$ is a smooth function generalizing the logistic function \begin{eqnarray*} f(u)=ru-bu^\gamma,\,\,\,\,\,\, u\geq0,\,\,\text{with}\,\, r\geq0,\,\,b>0\,\,\text{and}\,\,\gamma>1. \end{eqnarray*} It is proved that the corresponding initial-boundary value problem possesses a unique global classical solution that is uniformly bounded provided that some technical conditions are fulfilled.