Boundedness in a two-dimensional chemotaxis system with signal-dependent motility and logistic source

In this paper, we study the following chemotaxis system with a signal-dependent motility and logistic source: {u(t)= Delta(gamma(v)u)+mu u(1-u(alpha)), x is an element of Omega,t > 0, 0=Delta v-v+u(r), x is an element of Omega, t > 0, u(x,0)=u(0)(x), x is an element of Omega under homogeneous Neumann boundary conditions in a smooth bounded domain Omega is an element of R-2, where the motility function gamma(v) satisfies gamma(v)is an element of C-3([0,8 infinity)) with gamma(v)>0, and vertical bar gamma'(v)vertical bar(2) /gamma(v) is bounded for all v > 0. The purpose of this paper is to prove that the modelpossesses globally bounded solutions. In addition, we show that all solutions (u,v) of the model will exponentially converge to the unique constant steady state (1,1) as t ->+infinity when mu >= K/4(1+r) with K= max(0<v <=infinity)vertical bar gamma'(v)vertical bar(2)/gamma(v).