Global boundedness of solutions to a parabolic-parabolic chemotaxis system with local sensing in higher dimensions

This paper deals with classical solutions to the parabolic-parabolic system ut=Δ(Δ(v)u)inω×(0,∞),vt=Δv-v+uinω×(0,∞),∂u∂ν=∂v∂ν=0on∂ω×(0,∞),u(·,0)=u0,v(·,0)=v0inω, where ω is a smooth bounded domain in R n (n ∼ 3), Δ(v) = v -k (k > 0) and the initial data (u 0, v 0) is positive and regular. This system has striking features similar to those of the logarithmic Keller-Segel system. It is established that classical solutions of the system exist globally in time and remain uniformly bounded in time if k ϵ(0, n/(n - 2)), independently of the magnitude of mass. This constant n/(n - 2) is conjectured as the optimal range guaranteeing global existence and boundedness in the corresponding logarithmic Keller-Segel system. In the course of our analysis we introduce an auxiliary function and derive an evolution equation that it satisfies. Using refined comparison estimates we control the behavior of the nonlinear term in the equation for the auxiliary function, and this in turn yields sufficient information to control solutions of the original system.