Blow-up phenomena in a parabolic-elliptic-elliptic attraction-repulsion chemotaxis system with superlinear logistic degradation

Abstract

This paper is concerned with the attraction-repulsion chemotaxis system with superlinear logistic degradation, \begin{align*} \begin{cases} u_t = Δu - χ\nabla\cdot(u \nabla v) + ξ\nabla\cdot (u \nabla w) + λu - μu^k, \quad &x \in Ω,\ t>0,\\[1.05mm] 0= Δv + αu - βv, \quad &x \in Ω,\ t>0,\\[1.05mm] 0= Δw + γu - δw, \quad &x \in Ω,\ t>0, \end{cases} \end{align*} under homogeneous Neumann boundary conditions, in a ball $\Omega \subset {\mathbb{R}}^n$ ($n\ge 3$), with constant parameters $\lambda \in \mathbb{R}$, $k>1$, $\mu ,\chi ,\xi ,\alpha ,\beta ,\gamma ,\delta >0$. Blow-up phenomena in the system have been well investigated in the case $\lambda =\mu =0$, whereas the attraction-repulsion chemotaxis system with logistic degradation has been not studied. Under the condition that $k>1$ is close to $1$, this paper ensures a solution which blows up in $L^{\infty }$-norm and $L^{\sigma }$-norm with some $\sigma >1$ for some nonnegative initial data. Moreover, a lower bound of blow-up time is derived.

References in corpus (1)

• Blow-up in a quasilinear parabolic-elliptic Keller-Segel system with logistic source