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

2019/02/26 by Nishino, Teruto, Yokota, Tomomi
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1902.09787

Abstract

This paper deals with a lower bound for the blow-up time for solutions of the fully parabolic chemotaxis system \begincases u_t=∇ ⋅ [(u+α)^m₁-1 ∇ u-χu(u+α)^m₂-2 ∇ v] amp; \rm in Ω× (0,T),
v_t=Δv-v+u amp; \rm in Ω× (0,T) \endcases under Neumann boundary conditions and initial conditions, where Ω is a general bounded domain in ℝⁿ with smooth boundary, α>0, χ>0, m₁, m₂ ∈ ℝ and T>0. Recently, Anderson-Deng (2017) gave a lower bound for the blow-up time in the case that m₁=1 and Ω is a convex bounded domain. The purpose of this paper is to generalize the result in Anderson-Deng (2017) to the case that m₁ ≠ 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.

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