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# The fast signal diffusion limit in nonlinear chemotaxis systems

• 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 $p\in[1, \infty]$.

Mathematics Subject Classification: 92C17, 35K55, 35B40.

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