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July  2019, 24(7): 3357-3377. doi: 10.3934/dcdsb.2018324

## Global existence and boundedness of solution of a parabolic-parabolic-ODE chemotaxis-haptotaxis model with (generalized) logistic source

 1 Department of Basic Science, Jilin Jianzhu University, Changchun 130118, China 2 School of Information, Renmin University of China, Beijing, 100872, China 3 School of Mathematics and Statistics Science, Ludong University, Yantai 264025, China

Received  March 2018 Revised  August 2018 Published  January 2019

In this paper, we study the following chemotaxis–haptotaxis system with (generalized) logistic source
 $\left\{\begin{array}{ll} u_t = \Delta u-\chi\nabla\cdot(u\nabla v)- \xi\nabla\cdot(u\nabla w)+u(a-\mu u^{r-1}-w), \\ {v_t = \Delta v- v +u}, \quad \\ {w_t = - vw}, \quad\\ {\frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = \frac{\partial w}{\partial \nu} = 0}, \quad x\in \partial\Omega, t>0, \\ {u(x, 0) = u_0(x)}, v(x, 0) = v_0(x), w(x, 0) = w_0(x), \quad x\in \Omega, \ \end{array}\right. ~~~~~~~~~~~~~~~~~(0.1)$
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
 $\begin{equation*} \mu>\mu^{*} = \begin{array}{ll} \frac{(N-2)_{+}}{N}(\chi+C_{\beta}) C^{\frac{1}{\frac{N}{2}+1}}_{\frac{N}{2}+1}, if r = 2, \ \end{array} \end{equation*}$
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.
Citation: Ling Liu, Jiashan Zheng. Global existence and boundedness of solution of a parabolic-parabolic-ODE chemotaxis-haptotaxis model with (generalized) logistic source. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3357-3377. doi: 10.3934/dcdsb.2018324
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