On the optimality of upper estimates near blow-up in quasilinear Keller–Segel systems

ABSTRACT

Solutions $(u,v)$ to the chemotaxis system $\begin{array}{rl}& u_t=\mathrm{\nabla }\cdot \left(\right(u+1{)}^{m-1}\mathrm{\nabla }u-u(u+1{)}^{q-1}\mathrm{\nabla }v),\\ & \tau v_t=\mathrm{\Delta }v-v+u\end{array}$in a ball $\mathrm{\Omega }\subset {\mathbb{R}}^n$, $n\ge 2$, wherein $m,q\in \mathbb{R}$ and $\tau \in \{0,1\}$ are given parameters with m−q>−1, cannot blow up in finite time provided u is uniformly-in-time bounded in $L^p\left(\mathrm{\Omega }\right)$ for some $p>p_0:=\frac{n}{2}(1-(m-q\left)\right)$. For radially symmetric solutions, we show that, if u is only bounded in $L^{p_0}\left(\mathrm{\Omega }\right)$ and the technical condition $m>\frac{n-2p_0}{n}$ is fulfilled, then, for any $\alpha >\frac{n}{p_0}$, there is C>0 with $u(x,t)\le C|x{|}^{-\alpha }\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}x\in \mathrm{\Omega }\mathrm{a}\mathrm{n}\mathrm{d}t\in (0,T_{max}),$$T_{max}\in (0,\mathrm{\infty }]$ denoting the maximal existence time. This is essentially optimal in the sense that, if this estimate held for any $\alpha <\frac{n}{p_0}$, then u would already be bounded in $L^p\left(\mathrm{\Omega }\right)$ for some $p>p_0$.

Keywords:

• Blow-up profile
• nonlinear diffusion
• gradient estimates
• chemotaxis

AMS Classification (2020):

• Primary: 35B40
• Secondary: 35K40
• 35K65
• 92C17

Acknowledgments

The author is partially supported by the German Academic Scholarship Foundation and by the Deutsche Forschungsgemeinschaft within the project Emergence of structures and advantages in cross-diffusion systems, project number 411007140.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The author is partially supported by the Studienstiftung des Deutschen Volkes (German Academic Scholarship Foundation) and by the Deutsche Forschungsgemeinschaft within the project Emergence of structures and advantages in cross-diffusion systems, project number 411007140.