Boundedness and finite-time blow-up in a quasilinear parabolic-elliptic-elliptic attraction-repulsion chemotaxis system

Abstract

This paper deals with the quasilinear attraction-repulsion chemotaxis system \begin{align*} \begin{cases} u_t=\nabla\cdot \big((u+1)^{m-1}\nabla u -χu(u+1)^{p-2}\nabla v +ξu(u+1)^{q-2}\nabla w\big) +f(u), \\[1.05mm] 0=Δv+αu-βv, \\[1.05mm] 0=Δw+γu-δw \end{cases} \end{align*} in a bounded domain $\Omega \subset {\mathbb{R}}^n$ ($n\in \mathbb{N}$) with smooth boundary $\partial \Omega$, where $m,p,q\in \mathbb{R}$, $\chi ,\xi ,\alpha ,\beta ,\gamma ,\delta >0$ are constants. Moreover, it is supposed that the function $f$ satisfies $f(u)\equiv 0$ in the study of boundedness, whereas, when considering blow-up, it is assumed that $m>0$ and $f$ is a function of logistic type such as $f(u)=\lambda u-\mu u^{\kappa }$ with $\lambda \ge 0$, $\mu >0$ and $\kappa >1$ sufficiently close to~$1$, in the radially symmetric setting. In the case that $\xi =0$ and $f(u)\equiv 0$, global existence and boundedness have been proved under the condition $p<m+\frac{2}{n}$. Also, in the case that $m=1$, $p=q=2$ and $f$ is a function of logistic type, finite-time blow-up has been established by assuming $\chi \alpha -\xi \gamma >0$. This paper classifies boundedness and blow-up into the cases $p<q$ and $p>q$ without any condition for the sign of $\chi \alpha -\xi \gamma$ and the case $p=q$ with $\chi \alpha -\xi \gamma <0$ or $\chi \alpha -\xi \gamma >0$.

References in corpus (4)

• New critical exponents in a fully parabolic quasilinear Keller-Segel and applications to volume filling models
• Relaxed parameter conditions for chemotactic collapse in logistic-type parabolic-elliptic Keller-Segel systems
• Blow-up in a quasilinear parabolic-elliptic Keller-Segel system with logistic source
• Blow-up phenomena in a parabolic-elliptic-elliptic attraction-repulsion chemotaxis system with superlinear logistic degradation