A new (and optimal) result for boundedness of solution of a quasilinear chemotaxis--haptotaxis model (with logistic source)

This article deals with an initial-boundary value problem for the coupled chemotaxis-haptotaxis system with nonlinear diffusion $$\{\begin{array}{l}{u}_t=\mathrm{\nabla }\cdot (D(u)\mathrm{\nabla }u)-\chi \mathrm{\nabla }\cdot (u\mathrm{\nabla }v)-\xi \mathrm{\nabla }\cdot (u\mathrm{\nabla }w)+\mu u(1-u-w),x\in \mathrm{\Omega },t>0,\\ \tau v_t=\mathrm{\Delta }v-v+u,x\in \mathrm{\Omega },t>0,\\ w_t=-vw,x\in \mathrm{\Omega },t>0,\end{array}$$ under homogeneous Neumann boundary conditions in a smooth bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^N(N\ge 1)$, where $\tau \in \{0,1\}$ and $\chi$, $\xi$ and $\mu$ are given nonnegative parameters. As far as we know, this situation provides the first {\bf rigorous} result which (precisely) gives the relationship between $m,\xi ,\chi$ and $\mu$ that yields to the boundedness of the solutions. Moreover, these results thereby significantly extending results of previous results of several authors (see Remarks 1.1 and 1.2) and some optimal results are obtained.