# American Institute of Mathematical Sciences

June  2018, 23(4): 1675-1688. doi: 10.3934/dcdsb.2018069

## Boundedness and global solvability to a chemotaxis-haptotaxis model with slow and fast diffusion

 School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China

* Corresponding author: Chunhua Jin

Received  June 2017 Revised  August 2017 Published  June 2018 Early access  January 2018

Fund Project: The author is supported by NSFC(11471127), Guangdong Natural Science Funds for Distinguished Young Scholar (2015A030306029).

In this paper, we deal with the following coupled chemotaxis-haptotaxis system modeling cancer invasionwith nonlinear diffusion,
 $\left\{ \begin{array}{l}{u_t} = \Delta {u^m} - \chi \nabla \cdot \left( {u \cdot \nabla v} \right) - \xi \nabla \cdot \left( {u \cdot \nabla w} \right) + \mu u\left( {1 - u - w} \right),{\rm{in}}\;\Omega \times {{\mathbb{R}}^ + },\\{v_t} - \nabla v + v = u,\;{\rm{in}}\;\Omega \times {{\mathbb{R}}^ + },\\{w_t} = - vw,\;\;{\rm{in}}\;\Omega \times {{\mathbb{R}}^ + },\end{array} \right.$
where
 $Ω\subset\mathbb R^N$
(
 $N≥ 3$
) is a bounded domain. Under zero-flux boundary conditions, we showed that for any
 $m>0$
, the problem admits a global bounded weak solution for any large initial datum if
 $\frac{χ}{μ}$
is appropriately small. The slow diffusion case (
 $m>1$
) of this problem have been studied by many authors [14,7,19,23], in which, the boundedness and the global in time solution are established for
 $m>\frac{2N}{N+2}$
, but the cases
 $m≤ \frac{2N}{N+2}$
remain open.
Citation: Chunhua Jin. Boundedness and global solvability to a chemotaxis-haptotaxis model with slow and fast diffusion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1675-1688. doi: 10.3934/dcdsb.2018069
##### References:

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