Effects of density-suppressed motility in a two-dimensional chemotaxis model arising from tumor invasion

This paper deals with the global stability of the following density-suppressed motility system ut=Delta(phi (v)u),x is an element of Omega,t>0,vt=Delta v+wz,x is an element of Omega,t>0,wt=-wz,x is an element of Omega,t>0,zt=Delta z-z+u,x is an element of Omega,t>0in a bounded domain Omega subset of R2 with smooth boundary, where the motility function phi (v) is positive. If phi (v) has the lower-upper bound, we can obtain that this system possesses a unique bounded classical solution. Moreover, we can obtain that the global solution (u, v, w, z) will exponentially converge to the equilibrium (u0,v0+w0,0,u0) as t ->+infinity, where f0=1|Omega |integral Omega f0(x)dx.