Blow-Up Solution in Logarithmic Sensitivity and Indirect Signal Generation Keller–Segel System

Abstract

In this paper, we consider the following indirect signal generation and logarithmic sensitivity ε n t = Δ n − χ ∇ · n ∇ ln c x ∈ Ω , t > 0 c t = Δ c − c + w x ∈ Ω , t > 0 w t = Δ w − w + n x ∈ Ω , t > 0 under homogeneous Neumann boundary conditions in a ball domain Ω ⊂ R N N ≥ 4 with smooth boundary ∂ Ω . This paper considers in the singular limit ε ⟶ 0 ; the result comes from the finite time blow-up of arbitrary large values of n in the corresponding nonlocal scalar parabolic equation case when N ≥ 4 and χ > 2 N / N − 2 .