Stabilization in the Keller–Segel system with signal-dependent sensitivity

Abstract

This paper deals with the Keller–Segel system with signal-dependent sensitivity $\begin{array}{rl}& u_t=\mathrm{\Delta }u-\chi \mathrm{\nabla }\cdot \left(uS\right(v\left)\mathrm{\nabla }v\right),\\ & v_t=\mathrm{\Delta }v-v+u,\end{array}$ where $\chi >0$ and S is a given function generalizing the sensitivity $S\left(s\right)=1/(a+s{)}^k$, k>1, $a\ge 0$, and shows exponential convergence of global classical solutions under an additional smallness condition for $\chi >0$.

COMMUNICATED BY:

• J. Shi

Keywords:

• Chemotaxis system
• signal-dependent sensitivity
• asymptotic behaviour

MSC (2010):

• 35B40 (primary)
• 35K51
• 92C17
• 35Q92 (secondary)

Acknowledgements

The authors would like to thank the anonymous referees for their helpful comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

TB and JL acknowledge support of the Deutsche Forschungsgemeinschaft within the project Analysis of chemotactic cross-diffusion in complex frameworks. MM is funded by Japan Society for the Promotion of Science (JSPS) Research Fellowships for Young Scientists (No. 17J00101) and JSPS Overseas Challenge Programme for Young Researchers.