Boundedness of solution of a parabolic--ODE--parabolic chemotaxis--haptotaxis model with (generalized) logistic source

Abstract

In this paper, we study the following chemotaxis--haptotaxis system with (generalized) logistic source $$ \left\{\begin{array}{ll} u_t=\Delta u-\chi\nabla\cdot(u\nabla v)- \xi\nabla\cdot(u\nabla w)+u(a-\mu u^{r-1}-w), \displaystyle{v_t=\Delta v- v +u},\quad \\ \displaystyle{w_t=- vw},\quad\\ \displaystyle{\frac{\partial u}{\partial \nu}=\frac{\partial v}{\partial \nu}=\frac{\partial w}{\partial \nu}=0},\quad x\in \partial\Omega, t>0,\\ \displaystyle{u(x,0)=u_0(x)},v(x,0)=v_0(x),w(x,0)=w_0(x),\quad x\in \Omega, \end{array}\right.\eqno(0.1) $$ %under homogeneous Neumann boundary conditions in a smooth bounded domain $\mathbb{R}^N(N\geq1)$, with parameter $r>1$. the parameters $a\in \mathbb{R}, \mu>0, \chi>0$. It is shown that when $r>2$, or \begin{equation*} \mu>\mu^{*}=\begin{array}{ll} \frac{(N-2)_{+}}{N}(\chi+C_{\beta}) C^{\frac{1}{\frac{N}{2}+1}}_{\frac{N}{2}+1},~~~\mbox{if}~~r=2, \end{array} \end{equation*} % $\mu>\frac{(N-2)_{+}}{N}\chi C^{\frac{1}{\frac{N}{2}+1}}_{\frac{N}{2}+1}$, the considered problem possesses a global classical solution which is bounded, where $C^{\frac{1}{\frac{N}{2}+1}}_{\frac{N}{2}+1}$ is a positive constant which is corresponding to the maximal sobolev regularity. Here $C_{\beta}$ is a positive constant which depends on $\xi$, $\|u_0\|_{C(\bar{\Omega})},\|v_0\|_{W^{1,\infty}(\Omega)}$ and $\|w_0\|_{L^\infty(\Omega)}$. This result improves or extends previous results of several authors.