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\begin{document}

\title{Finite time collapse in chemotaxis systems with logistic-type
superlinear source}



\author[1]{Monica Marras}%
\author[2]{Stella Vernier-Piro}%
\affil[1]{University of Cagliari}%
\affil[2]{Universita di Cagliari}%


\vspace{-1em}



  
  \date{June 10, 2024}


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\selectlanguage{english}
\begin{abstract}
We consider the following quasilinear Keller-Segel system
\textbackslash{}begin\{equation*\}
\textbackslash{}left\textbackslash{}\{
\textbackslash{}begin\{array\}\{l\} \textbackslash{}begin\{aligned\}
u\_t = \textbackslash{}Delta u - \textbackslash{}nabla (u
\textbackslash{}nabla v) + g(u), \textbackslash{}quad
\&(x,t)\textbackslash{}in \textbackslash{}Omega \textbackslash{}times
{[}0,T\_\{max\}) ,\textbackslash{}\textbackslash{}{[}6pt{]} 0=
\textbackslash{}Delta v - v + u, \textbackslash{}qquad
\textbackslash{}quad \&(x,t)\textbackslash{}in \textbackslash{}Omega
\textbackslash{}times {[}0,T\_\{max\}),
\textbackslash{}\textbackslash{}{[}6pt{]} \textbackslash{}end\{aligned\}
\textbackslash{}end\{array\} \textbackslash{}right.
\textbackslash{}end\{equation*\} on a ball \$\textbackslash{}Omega
\textbackslash{}equiv
B\_R(0)\textbackslash{}subset\textbackslash{}mathbb\{R\}\^{}n\$,
\$n\textbackslash{}geq 3\$, \$R\textgreater{}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)=\textbackslash{}lambda u - \textbackslash{}mu
u\^{}k,\textbackslash{} k\textgreater{}1, \textbackslash{}
\textbackslash{}mu \textgreater{}0\$, \$\textbackslash{}lambda
\textbackslash{}in \textbackslash{}mathbb\{R\}\$ and \$T\_\{max\}\$ is
the blow-up time.\textbackslash{}\textbackslash{} 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\^{}\{\textbackslash{}infty\} (\textbackslash{}Omega)\$-norm
\textbackslash{}cite\{W\}, blows up also in
\$L\^{}p(\textbackslash{}Omega)\$-norm for some \$p\textgreater{}1\$.
Moreover a lower bound of the lifespan (or blow-up time when it is
finite) \$T\_\{max\}\$ is derived. \textbackslash{}\textbackslash{} In
addition, if \$\textbackslash{}Omega \textbackslash{}subset
\textbackslash{}mathbb\{R\}\^{}3\$ a lower bound of \$T\_\{max\}\$ is
explicitly computable.%
\end{abstract}%



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