Global existence, boundedness and stabilization in a high-dimensional chemotaxis system with consumption

This paper deals with the homogeneous Neumann boundary-value problem for the chemotaxis-consumption system \ align & {{uₓ}= u- (u v) + u- {u^2}, \ \ \ \ \ \ \ x, t>0, \\ & {vₓ}= v-uv, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x, t>0, \\ align. in N-dimensional bounded smooth domains for suitably regular positive initial data. We shall establish the existence of a global bounded classical solution for suitably large μ and prove that for any μ>0 there exists a weak solution. Moreover, in the case of κ>0 convergence to the constant equilibrium (κμ, 0) is shown.