Finite time collapse in chemotaxis systems with logistic-type
superlinear source

- Monica Marras,
- Stella Vernier-Piro

## Abstract

We consider the following quasilinear Keller-Segel system
\begin{equation*}
\left\{
\begin{array}{l} \begin{aligned}
u_t = \Delta u - \nabla (u
\nabla v) + g(u), \quad
&(x,t)\in \Omega \times
[0,T_{max}) ,\\[6pt] 0=
\Delta v - v + u, \qquad
\quad &(x,t)\in \Omega
\times [0,T_{max}),
\\[6pt] \end{aligned}
\end{array} \right.
\end{equation*} on a ball $\Omega
\equiv
B_R(0)\subset\mathbb{R}^n$,
$n\geq 3$, $R>0$, under homogeneous
Neumann boundary conditions and non negative initial data. The source
term $g(u)$ is superlinear and of logistic type i.e.
$g(u)=\lambda u - \mu
u^k,\ k>1, \
\mu >0$, $\lambda
\in \mathbb{R}$ and $T_{max}$ is
the blow-up time.\\ The solution $(u,v)$
may or may not blow up in finite time. Under suitable conditions on
data, we prove that the function $u$, which blows up in
$L^{\infty} (\Omega)$-norm
\cite{W}, blows up also in
$L^p(\Omega)$-norm for some $p>1$.
Moreover a lower bound of the lifespan (or blow-up time when it is
finite) $T_{max}$ is derived. \\ In
addition, if $\Omega \subset
\mathbb{R}^3$ a lower bound of $T_{max}$ is
explicitly computable.

24 Feb 2020Submitted to *Mathematical Methods in the Applied Sciences* 28 Feb 2020Submission Checks Completed

28 Feb 2020Assigned to Editor

04 Mar 2020Reviewer(s) Assigned

16 Jun 2020Review(s) Completed, Editorial Evaluation Pending

16 Jun 2020Editorial Decision: Revise Minor

22 Jun 20201st Revision Received

23 Jun 2020Submission Checks Completed

23 Jun 2020Assigned to Editor

23 Jun 2020Editorial Decision: Accept