Abstract:

This paper is concerned with an initial-boundary value problem for the following chemotaxis-haptotaxis model $\left\{ \begin{array}{ll} u_{t}=\nabla\cdot (D(u) \nabla u)-\chi \nabla\cdot (\frac{u}{(1+u)^{\alpha}}\nabla v)-\xi \nabla\cdot (\frac{u}{(1+u)^{\beta}}\nabla w)+ u(a-\mu u^{k-1}-\lambda w),\\ v_{t}=\triangle v-v+u^{\gamma}, x\in \Omega,\;t>0,\\ w_{t}=-v w, x\in \Omega,\;t>0, \end{array}\right. $ under homogenous Neumann boundary condition in a bounded domain $ \Omega \subset \mathbb{R} ^{3} $, with $ \chi, \xi, \mu,\lambda,$ $\gamma >0$, $k>1$, $a \in \mathbb{R} $, and $D(u)\geq C_{D} (u+1)^{m-1}$ for $C_{D}>0, m\in \mathbb{R} $. It is shown that(i) For $0<\gamma\leq\frac{2}{3}$, if $\alpha>\gamma-k+1$ and $\beta>1-k$, there is a classical solution $(u, v, w)$ which is globally bounded to the above problem.(ii) For $\frac{2}{3}<\gamma\leq1$, if $\alpha>\gamma-k+\frac{1}{e}+1$ and $\beta>\max\{\frac{(3\gamma-2)(3\gamma+2k-2)}{6}-k+1, \frac{(3\gamma-2)(\gamma+\frac{1}{e})}{3}-k+1\}, $ or $\alpha>\gamma-k+1$ and $\beta>\max\{\frac{(3\gamma-2)(3\gamma+2k-2)}{6}-k+1, \frac{(3\gamma-2) (\alpha+k-1)}{3}-k+1\}, $ there is a classical solution $(u, v, w)$ which is globally bounded to the above problem.

Key words: Global existence, Boundedness, Chemotaxis-haptotaxis, Nonlinear diffusion