Explicit lower bound of blow-up time in a fully parabolic chemotaxis system with nonlinear cross- diffusion

This paper detects the lower bounds of blow-up time of smooth solutions for the chemotaxis model \bas \left\{ \begin{array}{ll} u_t= \Delta u - \chi\nabla \cdot (u(u+1)^{m-1}\nabla v), \qquad & x\in B_1(0), \, t>0, \\[1mm] v_t=\Delta v-v+u, \qquad &x\in B_1(0), \, t>0, \end{array} \right. \eas under homogeneous Neumann boundary conditions in a unit ball $B_1(0) \subset \R^3$ centered at the origin, with positive constants $\chi$ and parameter $m\in\R$.\abs % Under the assumption that $(u(x, 0), v(x, 0))=(u_0(|x|), v_0(|x|)) \in C^0(\bar{B}_1(0)) \times W^{1, \infty}(B_1(0))$, it is shown that whenever $m\in [\frac{2}{3}, 2]$, the blow-up time of a classical solution to the corresponding initial-boundary problem has an explicit lower bound measured in terms of $\chi$, $\int_{B_1(0)} u_0^p$ and $\int_{B_1(0)} |\nabla v_0|^{2q}$ for appropriate $p>1$ and $q>1$. Here we underline that the global classical solution exists and is bounded if $m<\frac{2}{3}$, which leads to the assumption $m\ge \frac{2}{3}$ for addressing the properties of blow-up solutions. However, the question of lower bounds of blow-up time for the case $m>2$ remains open due to technical reasons.