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 $\left\{\begin{array}{cc}\epsilon n_t=\mathrm{\Delta }n-\chi \nabla ·\left(n\nabla \ln c\right)& x\in \mathrm{\Omega },t>0\\ c_t=\mathrm{\Delta }c-c+w& x\in \mathrm{\Omega },t>0\\ w_t=\mathrm{\Delta }w-w+n& x\in \mathrm{\Omega },t>0\end{array}\right.$ under homogeneous Neumann boundary conditions in a ball domain $\mathrm{\Omega }\subset R^N\left(N\ge 4\right)$ with smooth boundary $\partial \mathrm{\Omega }$ . This paper considers in the singular limit $\epsilon ⟶0$ ; the result comes from the finite time blow-up of arbitrary large values of $n$ in the corresponding nonlocal scalar parabolic equation case when $N\ge 4$ and $\chi >2N/N-2$ .