In this talk, we focus on the following chemotaxis system with signal-dependent motility

\begin{align*}

u_{\varepsilon t}&=\Delta(u_\varepsilon e^{-v_\varepsilon})+\varepsilon(\kappa u_\varepsilon-\mu u_\varepsilon^2),\\

0&=\Delta v_\varepsilon -v_\varepsilon+u_\varepsilon

\end{align*}

in a smooth bounded domain $\Omega\in\mathbb{R}^2$ with homogenuous Neumann boundary conditions and parameters $\kappa,\mu>0$ and $\varepsilon\in (0,1)$. It is known that classical solutions are global and bounded. Our aim is to find uniform convergence of solutions $(u_\varepsilon,v_\varepsilon)$ to solutions $(u,v)$ in the limit $\varepsilon\searrow 0$. The main challenge will be to derive $\varepsilon$-independent $L^p$-bounds for all $p\geq 1$, which will be local in time. \\

As a consequence, we can show that for certain initial data $u_0$ the solutions $u_\varepsilon$ exceed any certain value $M>0$ before a time $T(M)>0$ as long as $\varepsilon\in(0,\tilde{\varepsilon}(M))$ for some $\tilde{\varepsilon}(M)>0$. That result is based on the known infinite time blow-up in the $\varepsilon=0$-system that can occur, if $\Omega=B_R(0)$ for some $R>0$ and the initial data $u_0$ of supercritical mass is radially symmetric.