The fast signal diffusion limit in nonlinear chemotaxis systems.

For n≥2 let Ω⊂Rn be a bounded domain with smooth boundary as well as some nonnegative functions 0≢u0∈W1,∞(Ω) and v0∈W1,∞(Ω). With ε∈(0,1) we want to know in which sense (if any!) solutions to the parabolic-parabolic system⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩ut=∇⋅((u+1)m−1∇u)−∇⋅(u∇v)in Ω×(0,∞),εvt=Δv−v+uin Ω×(0,∞),∂u∂ν=∂v∂ν=0on ∂Ω×(0,∞),u(⋅,0)=u0, v(⋅,0)=v0in Ωconverge to those of the system where ε=0 and where the initial condition for v has been removed. We will see in our theorem that indeed the solutions of these systems converge in a meaningful way if m>1+n−2n without the need for further conditions, e. g. on the size of ∥u0∥Lp(Ω) for some p∈[1,∞]. [ABSTRACT FROM AUTHOR]