# 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

 Institut für Mathematik, Universität Paderborn, 33098 Paderborn, Germany

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:

show all references

##### References:
 [1] Hao Yu, Wei Wang, Sining Zheng. Boundedness of solutions to a fully parabolic Keller-Segel system with nonlinear sensitivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1635-1644. doi: 10.3934/dcdsb.2017078 [2] Kentarou Fujie, Takasi Senba. Global existence and boundedness in a parabolic-elliptic Keller-Segel system with general sensitivity. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 81-102. doi: 10.3934/dcdsb.2016.21.81 [3] Hao Yu, Wei Wang, Sining Zheng. Global boundedness of solutions to a Keller-Segel system with nonlinear sensitivity. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1317-1327. doi: 10.3934/dcdsb.2016.21.1317 [4] Xie Li, Zhaoyin Xiang. Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3503-3531. doi: 10.3934/dcds.2015.35.3503 [5] Miaoqing Tian, Sining Zheng. Global boundedness versus finite-time blow-up of solutions to a quasilinear fully parabolic Keller-Segel system of two species. Communications on Pure & Applied Analysis, 2016, 15 (1) : 243-260. doi: 10.3934/cpaa.2016.15.243 [6] Kentarou Fujie, Chihiro Nishiyama, Tomomi Yokota. Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with the sensitivity $v^{-1}S(u)$. Conference Publications, 2015, 2015 (special) : 464-472. doi: 10.3934/proc.2015.0464 [7] Sachiko Ishida, Tomomi Yokota. Boundedness in a quasilinear fully parabolic Keller-Segel system via maximal Sobolev regularity. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 211-232. doi: 10.3934/dcdss.2020012 [8] Marco Di Francesco, Donatella Donatelli. Singular convergence of nonlinear hyperbolic chemotaxis systems to Keller-Segel type models. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 79-100. doi: 10.3934/dcdsb.2010.13.79 [9] Qi Wang, Jingyue Yang, Feng Yu. Boundedness in logistic Keller-Segel models with nonlinear diffusion and sensitivity functions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 5021-5036. doi: 10.3934/dcds.2017216 [10] Piotr Biler, Ignacio Guerra, Grzegorz Karch. Large global-in-time solutions of the parabolic-parabolic Keller-Segel system on the plane. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2117-2126. doi: 10.3934/cpaa.2015.14.2117 [11] Mengyao Ding, Xiangdong Zhao. $L^\sigma$-measure criteria for boundedness in a quasilinear parabolic-parabolic Keller-Segel system with supercritical sensitivity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5297-5315. doi: 10.3934/dcdsb.2019059 [12] Jan Burczak, Rafael Granero-Belinchón. Boundedness and homogeneous asymptotics for a fractional logistic Keller-Segel equations. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 139-164. doi: 10.3934/dcdss.2020008 [13] Qi Wang. Boundary spikes of a Keller-Segel chemotaxis system with saturated logarithmic sensitivity. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1231-1250. doi: 10.3934/dcdsb.2015.20.1231 [14] Tobias Black. Global generalized solutions to a parabolic-elliptic Keller-Segel system with singular sensitivity. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 119-137. doi: 10.3934/dcdss.2020007 [15] Mengyao Ding, Sining Zheng. $L^γ$-measure criteria for boundedness in a quasilinear parabolic-elliptic Keller-Segel system with supercritical sensitivity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 2971-2988. doi: 10.3934/dcdsb.2018295 [16] Chao Deng, Tong Li. Global existence and large time behavior of a 2D Keller-Segel system in logarithmic Lebesgue spaces. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 183-195. doi: 10.3934/dcdsb.2018093 [17] Jinhuan Wang, Li Chen, Liang Hong. Parabolic elliptic type Keller-Segel system on the whole space case. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1061-1084. doi: 10.3934/dcds.2016.36.1061 [18] Wenting Cong, Jian-Guo Liu. Uniform $L^{∞}$ boundedness for a degenerate parabolic-parabolic Keller-Segel model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 307-338. doi: 10.3934/dcdsb.2017015 [19] Luca Battaglia. A general existence result for stationary solutions to the Keller-Segel system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 905-926. doi: 10.3934/dcds.2019038 [20] Hai-Yang Jin, Zhi-An Wang. Global stabilization of the full attraction-repulsion Keller-Segel system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3509-3527. doi: 10.3934/dcds.2020027

2019 Impact Factor: 1.27