Threshold value for a quasilinear Keller–Segel chemotaxis system with the intermediate exponent in a bounded domain

Abstract

We consider a quasilinear chemotaxis model $\left(\begin{array}{c}{u}_t=\nabla \cdot (D\left(u\right)\nabla u)-\nabla \cdot (S\left(u\right)\nabla v),\tau v_t=\Delta v-v+u,\end{array}\right)$ with nonlinear diffusion function $D\in C^2\left([0,\infty )\right)$ and chemotactic sensitivity $S\in C^2\left([0,\infty )\right)$ in a bounded domain $\Omega \subset R^d$ $(d\ge 3)$. Here the rate $D\left(s\right)/S\left(s\right)$ grows like $s^{2-m}$ with $2d/(d+2)<m<2-2/d$ as $s\to \infty$, and $\tau =0,1$.

It is first shown that there exists a $M_{\ast }>0$ such that if free energy with initial data is suitably small and ${\Vert u_0\Vert }_{L^1\left(\Omega \right)}^{\alpha }{\Vert u_0\Vert }_{L^m\left(\Omega \right)}^{\beta }<M_{\ast }$ with $\alpha =2/(2-m)-d/m>0$ and $\beta =d-2/(2-m)>0$, then the classical solutions to the above system are uniformly-in-time bounded. Second, in radially symmetric settings we can find $M^{\ast }>0$ such that ${\Vert u_0\Vert }_{L^1\left(\Omega \right)}^{\alpha }{\Vert u_0\Vert }_{L^m\left(\Omega \right)}^{\beta }>M^{\ast }$ and the corresponding solution must be unbounded. These results show that the global behavior of classical solutions is classified by the combination of norms of initial data when $2d/(d+2)<m<2-2/d$.