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