# American Institute of Mathematical Sciences

June  2017, 10(3): 395-412. doi: 10.3934/dcdss.2017019

## Traveling wave solutions for a one dimensional model of cell-to-cell adhesion and diffusion with monostable reaction term

 1 College of Mathematics, Jilin University, Changchun, Jilin 130012, China 2 Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA

* Corresponding author

Received  December 2015 Revised  November 2016 Published  February 2017

Fund Project: The work of L. Bao is sponsored by SRF for ROCS, SEM., the Twelfth Five-Year Plan project of Jilin Province's Educational Science.

This work is concerned with the properties of the traveling wave solutions of a one dimensional model of cell diffusion and aggregation, incorporating volume filling and cell-to-cell adhesion with net birth term
 $\begin{equation*} ρ_t = [ D(ρ)ρ_x]_x + g(ρ)\ \ \ t≥0,\ \ \ x∈ \mathbb{R},\end{equation*}$
where
 $D(ρ)$
may take positive or negative values with different population density
 $ρ$
 $α ∈ [0,1]$
, and the negative one will lead to the ill-posedness of the equation. In all these cases we prove the existence of infinitely many traveling wave solutions, where these solutions are parameterized by their wave speed and monotonically connect the stationary states
 $ρ\equiv0$
and
 $ρ\equiv 1$
.
Citation: Lianzhang Bao, Zhengfang Zhou. Traveling wave solutions for a one dimensional model of cell-to-cell adhesion and diffusion with monostable reaction term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 395-412. doi: 10.3934/dcdss.2017019
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