Effect of nonlinear diffusion on a lower bound for the blow-up time in a fully parabolic chemotaxis system.

This paper deals with a lower bound for the blow-up time for solutions of the fully parabolic chemotaxis system { u t = ∇ ⋅ [ (u + α) m 1 − 1 ∇ u − χ u (u + α) m 2 − 2 ∇ v ] in Ω × (0 , T) , v t = Δ v − v + u in Ω × (0 , T) under Neumann boundary conditions and initial conditions, where Ω is a general bounded domain in R n with smooth boundary, α > 0 , χ > 0 , m 1 , m 2 ∈ R and T > 0. Recently, Anderson–Deng [1] gave a lower bound for the blow-up time in the case that m 1 = 1 and Ω is a convex bounded domain. The purpose of this paper is to generalize the result in [1] to the case that m 1 ≠ 1 and Ω is a non-convex bounded domain. The key to the proof is to make a sharp estimate by using the Gagliardo–Nirenberg inequality and an inequality for boundary integrals. As a consequence, the main result of this paper reflects the effect of nonlinear diffusion and need not assume the convexity of Ω. [ABSTRACT FROM AUTHOR]