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] $.