Critical mass phenomena in higher dimensional quasilinear Keller-Segel systems with indirect signal production

Abstract

In this paper, we deal with quasilinear Keller--Segel systems with indirect signal production, $$\{\begin{array}{ll}{u}_t=\nabla \cdot ((u+1{)}^{m-1}\nabla u)-\nabla \cdot (u\nabla v),& x\in \Omega ,t>0,\\ 0=\Delta v-\mu (t)+w,& x\in \Omega ,t>0,\\ w_t+w=u,& x\in \Omega ,t>0,\end{array}$$ complemented with homogeneous Neumann boundary conditions and suitable initial conditions, where $\Omega \subset {\mathbb{R}}^n$ $(n\ge 3)$ is a bounded smooth domain, $m\ge 1$ and $$μ(t) := \frac{1}{|Ω|} w(\cdot, t) \qquad\mbox{for}\ t>0.$$ We show that in the case $m\ge 2-\frac{2}{n}$, there exists $M_c>0$ such that if either $m>2-\frac{2}{n}$ or ${\int }_{\Omega }{u}_0<M_c$, then the solution exists globally and remains bounded, and that in the case $m\le 2-\frac{2}{n}$, if either $m<2-\frac{2}{n}$ or $M>2^{\frac{n}{2}}{n}^{n-1}{\omega }_n$, then there exist radially symmetric initial data such that ${\int }_{\Omega }{u}_0=M$ and the solution blows up in finite or infinite time, where the blow-up time is infinite if $m=2-\frac{2}{n}$. In particular, if $m=2-\frac{2}{n}$ there is a critical mass phenomenon in the sense that $\inf \left\{M>0:\exists u_0\text{ with }{\int }_{\Omega }{u}_0=M\text{ such that the corresponding solution blows up in infinite time}\right\}$ is a finite positive number.

18 pages

References in corpus (1)

• New critical exponents in a fully parabolic quasilinear Keller-Segel and applications to volume filling models