Persistence of mass in a chemotaxis system with logistic source

This paper studies the dynamical properties of the chemotaxis system {u(t)=Delta u-del.(u del v)+ru-mu u(2), x epsilon Omega, t>0, {v(t) =Delta v - v +u, x epsilon Omega, t>0, under homogeneous Neumann boundary conditions in bounded convex domains Omega subset of R-n, n >= 1, with positive constants x, r and mu. Numerical simulations but also some rigorous evidence have shown that depending on the relative size of r, mu and vertical bar Omega vertical bar, in comparison to the well-understood case when x = 0, this problem may exhibit quite a complex solution behavior, including unexpected effects such as asymptotic decay of the quantity u within large subdomains of Omega. The present work indicates that any such extinction phenomenon, if occurring at all, necessarily must be of spatially local nature, whereas the population as a whole always persists. More precisely, it is shown that for any nonnegative global classical solution (u, v) of (*) with u not equal 0 one can find m*. > 0 such that