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This article deals with an initial-boundary value problem for the coupledchemotaxis-haptotaxis system with nonlinear diffusion$$\left\{\begin{array}{ll} u_t=\nabla\cdot( D(u)\nablau)-\chi\nabla\cdot(u\nabla v)- \xi\nabla\cdot(u\nabla w)+\mu u(1- u-w), x\in\Omega, t>0,\\ \tau v_t=\Delta v- v +u,\quad x\in \Omega, t>0,\\ w_t=- vw,\quadx\in \Omega, t>0, \end{array}\right.$$ under homogeneous Neumann boundaryconditions in a smooth bounded domain $\Omega\subset\mathbb{R}^N(N\geq1)$,where $\tau\in\{0,1\}$ and $\chi$, $\xi$ and $\mu$ are given nonnegativeparameters. 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 resultsthereby significantly extending results of previous results of several authors(see Remarks 1.1 and 1.2) and some optimal results are obtained.

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