Large time behavior of solutions to a chemotaxis model with porous medium diffusion

In this paper, we study the large time behavior of a chemotaxis model with nonlinear diffusion and consumption { u t = Δ u m − ∇ ⋅ ( u ∇ v ) + μ u ( 1 − u ) , v t − Δ v = − v u , where m > 1 . In a previous paper [5] , we have proved the existence and uniform boundedness of global weak solutions for any nonnegative initial data and any m > 1 . In this work, we show that the weak solutions strongly converge to ( 1 , 0 ) in the large time limit.