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{ut=Δu−χ∇⋅(u(u+1)m−1∇v),x∈B1(0),t>0,vt=Δv−v+u,x∈B1(0),t>0, under homogeneous Neumann boundary conditions in a unit ball B1(0)⊂R3 centered at the origin, with positive constant χ and parameter m∈R. Under the assumption that (u(x,0),v(x,0))=(u0(|x|),v0(|x|))∈C0(B¯1(0))×W1,∞(B1(0)), it is shown that whenever m∈[23,2], the blow-up time of a classical solution to the corresponding initial-boundary problem has an explicit lower bound measured in terms of χ, ∫B1(0)u0p and ∫B1(0)|∇v0|2q for appropriate p>1 and q>1. Here we underline that the global classical solution exists and is bounded if m<23, which leads to the assumption m≥23 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.