| Abstract: |
| In this talk, we investigate the Neumann initial-boundary value problem for the following nonlinear chemotaxis system with indirect signal production:
\begin{align}\label{0}\tag{$\star$}
\begin{cases}
u_t = \Delta u - \nabla \cdot\left(f(u) \nabla v\right), \
0 = \Delta v - \mu(t) + w, \
w_t + w = u
\end{cases}
\end{align}
in $\Omega \subset \mathbb{R}^n$ for $n \geq 2$. Here, $\mu(t) := \fint_{\Omega} w(x, t) \,\mathrm{d}x$ and $f \in C^2([0,\infty))$ is a nonnegative function. We establish the following results:
\begin{itemize}
\item If $\Omega = B_R(0)$ for some $R > 0$ and $f(s) \geq k s^p$ for all $s \geq 1$ with constants $k > 0$ and $p > \frac{2}{n}$, then there exist radially symmetric initial data for which the corresponding solution blows up in finite time, regardless of the mass level $m := \int_{\Omega} u_0 \,\mathrm{d}x > 0$.
\item If $f(s) \leq K (s + 1)^p$ for all $s \geq 0$ with constants $K > 0$ and $p < \frac{2}{n}$, then for any appropriately regular initial data, the corresponding solution exists globally and remains bounded.
\end{itemize}
Our findings extend the results of Tao and Winkler (2017) regarding blow-up phenomena for \eqref{0} with $f(u) = u$ in the two-dimensional setting. |
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