Boundedness and global existence in a higher-dimensional parabolic-elliptic-ODE chemotaxis-haptotaxis model with remodeling of non-diffusible attractant

Abstract

This paper addresses the issue of boundedness for solutions to the following quasilinear chemotaxis-haptotaxis model of parabolic-elliptic-ODE type:$\{\begin{array}{c}{u}_t=\Delta u-\chi \nabla \cdot \left(u\nabla v\right)-\xi \nabla \cdot \left(u\nabla w\right)+u(r-\mu u^{\gamma -1}-w),x\in \Omega ,t>0,\\ 0=\Delta v+u-v,x\in \Omega ,t>0,\\ w_t=-vw+\eta w(1-u-w),x\in \Omega ,t>0,\end{array}$ subject to zero-flux boundary conditions within a smooth, bounded domain $\Omega \subset R^N$ (with $N\ge 3$). The parameters involved are $\chi >0,\mu >0,r\ge 0$, and $\eta >0$. It is demonstrated that, provided $\gamma >3-\frac{2}{N}$, for sufficiently smooth initial data, the corresponding initial-boundary problem admits a unique global-in-time classical solution, which remains uniformly bounded. To the best of our knowledge, these are the first results concerning the boundedness of solutions for this parabolic-elliptic-ODE system in higher dimensions.