# A refined criterion and lower bounds for the blow--up time in a
parabolic--elliptic chemotaxis system with nonlinear diffusion

Research paper by **Monica Marras, Teruto Nishino, Giuseppe Viglialoro**

Indexed on: **27 Jun '19**Published on: **08 Apr '19**Published in: **arXiv - Mathematics - Analysis of PDEs**

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#### Abstract

This paper deals with unbounded solutions to the following zero--flux
chemotaxis system \begin{equation}\label{ProblemAbstract} \tag{$\Diamond$}
\begin{cases}
% about u
u_t=\nabla \cdot [(u+\alpha)^{m_1-1}
\nabla u-\chi u(u+\alpha)^{m_2-2}
\nabla v]
&
(x,t) \in \Omega \times (0,T_{max}),
\\[1mm]
% about v
0=\Delta v-M+u
&
(x,t) \in \Omega \times (0,T_{max}),
\end{cases} \end{equation} where $\alpha>0$, $\Omega$ is a smooth and bounded
domain of $\mathbb{R}^n$, with $n\geq 1$, $t\in (0, T_{max})$, where $T_{max}$
the blow-up time, and $m_1,m_2$ real numbers. Given a sufficiently smooth
initial data $u_0:=u(x,0)\geq 0$ and set
$M:=\frac{1}{|\Omega|}\int_{\Omega}u_0(x)\,dx$, from the literature it is known
that under a proper interplay between the above parameters $m_1,m_2$ and the
extra condition $\int_\Omega v(x,t)dx=0$, system \eqref{ProblemAbstract}
possesses for any $\chi>0$ a unique classical solution which becomes unbounded
at $t\nearrow T_{max}$. In this investigation we first show that for
$p_0>\frac{n}{2}(m_2-m_1)$ any blowing up classical solution in
$L^\infty(\Omega)$--norm blows up also in $L^{p_0}(\Omega)$--norm. Then we
estimate the blow--up time $T_{max}$ providing a lower bound $T$.