Persistence of mass in a chemotaxis system with logistic source

Abstract

This paper studies the dynamical properties of the chemotaxis system(⋆){ut=Δu−χ∇⋅(u∇v)+ru−μu2,x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0, under homogeneous Neumann boundary conditions in bounded convex domains Ω⊂Rn, n≥1, with positive constants χ, r and μ.Numerical simulations but also some rigorous evidence have shown that depending on the relative size of r, μ and |Ω|, in comparison to the well-understood case when χ=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 Ω.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≢0 one can find m⋆>0 such that∫Ωu(x,t)dx≥m⋆for all t>0. The proof is based on an, in this context, apparently novel analysis of the functional ∫Ωln⁡u, deriving a lower bound for this quantity along a suitable sequence of times by appropriately exploiting a differential inequality for a suitable linear combination of ∫Ωln⁡u, ∫Ωu and ∫Ωv2.