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

arXiv admin note: text overlap with arXiv:1711.10044

Cited by in corpus (1)

• A new result for 2D boundedness of solutions to a chemotaxis--haptotaxis model with/without sub-logistic source