Boundedness in a three-dimensional chemotaxis-haptotaxis model

This paper studies the chemotaxis-haptotaxis system \begin{equation}\nonumber \left\{ \begin{array}{llc} u_t=Δu-χ\nabla\cdot(u\nabla v)-ξ\nabla\cdot(u\nabla w)+μu(1-u-w), &(x,t)\in Ω\times (0,T),\\ v_t=Δv-v+u, &(x,t)\inΩ\times (0,T),\\ w_t=-vw,&(x,t)\in Ω\times (0,T) \end{array} \right.\quad\quad(\star) \end{equation} under Neumann boundary conditions. Here $Ω\subset\mathbb{R}^3$ is a bounded domain with smooth boundary and the parameters $ξ,χ,μ>0$. We prove that for nonnegative and suitably smooth initial data $(u_0,v_0,w_0)$, if $χ/μ$ is sufficiently small, ($\star$) possesses a global classical solution which is bounded in $Ω\times(0,\infty)$. We underline that the result fully parallels the corresponding parabolic-elliptic-ODE system.

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