Global existence and uniform boundedness of smooth solutions to a parabolic-parabolic chemotaxis system with nonlinear diffusion
Global existence and uniform boundedness of smooth solutions to a parabolic-parabolic chemotaxis system with nonlinear diffusion
This paper is devoted to the following quasilinear chemotaxis system: $\bigl\{\scriptsize{ \begin{array}{l} u_{t}=\nabla\cdot(D(u)\nabla u)-\nabla\cdot(u\chi(v)\nabla v)+uf(u),\quad x\in\Omega,t>0, \\ v_{t}=\Delta v-ug(v),\quad x\in\Omega,t>0, \end{array} }\bigr. $ under homogeneous Neumann boundary conditions in a smooth bounded domain $\Omega\subset\mathbb{R}^{N}$ . The given functions $D(s)$ , $\chi(s)$ , $g(s)$ , and $f(s)$ are assumed to be …