Boundedness in a Chemotaxis-Consumption System with Singular Sensitivity

In this paper, we consider a fully parabolic chemotaxis system u(t) =Delta u - chi del. (u/v del v), v(t) = Delta v -lambda uv + mu v(1 - v) with singular sensitivity chi/v (chi > 0) on a boundary conditions Omega subset of R-n (n >= 2). On the assumption of chi is an element of (0, root 2/n), we obtain that the model possesses a unique global bounded classical solution. Moreover, under appropriate initial conditions, we prove parallel to v (., t)parallel to (L infinity(Omega)) -> 0 as t -> infinity.