# American Institute of Mathematical Sciences

March  2020, 25(3): 1109-1128. doi: 10.3934/dcdsb.2019211

## The fast signal diffusion limit in nonlinear chemotaxis systems

Received  November 2018 Revised  May 2019 Published  September 2019

For
 $n\geq2$
let
 $\mathit{\Omega }\subset {\mathbb{R}}^n$
be a bounded domain with smooth boundary as well as some nonnegative functions
 $0\not \equiv u_0\in W^{1, \infty}(\mathit{\Omega })$
and
 $v_0\in W^{1, \infty}(\mathit{\Omega })$
. With
 $\varepsilon\in(0, 1)$
we want to know in which sense (if any!) solutions to the parabolic-parabolic system
 $\begin{equation*} \begin{cases} u_t = \nabla\cdot((u+1)^{m-1}\nabla u)-\nabla \cdot(u\nabla v) \;\;\; & \text{in} \ \mathit{\Omega }\times\left(0, \infty \right), \\ \varepsilon v_t = \mathit{\Delta } v -v+u & \text{in} \ \mathit{\Omega }\times\left(0, \infty \right), \\ \frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = 0 & \text{on} \ \partial\mathit{\Omega }\times\left(0, \infty \right), \\ u(\cdot, 0) = u_0, \ v(\cdot, 0) = v_0 & \text{in} \ \mathit{\Omega } \end{cases} \end{equation*}$
converge to those of the system where
 $\varepsilon = 0$
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
 $m>1+\frac{n-2}{n}$
without the need for further conditions, e. g. on the size of
 $\left\|{{u_0}}\right\|_{L^p(\mathit{\Omega })}$
for some
 $p\in[1, \infty]$
.
Citation: Marcel Freitag. The fast signal diffusion limit in nonlinear chemotaxis systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1109-1128. doi: 10.3934/dcdsb.2019211
##### References:

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