Boundedness of classical solutions for a chemotaxis system with general sensitivity function

ABSTRACT

This paper deals with the chemotaxis system with general sensitivity function: $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{u}_t=\mathrm{\nabla }·(\mathit{\delta }\mathrm{\nabla }u-u\mathit{\chi }(v)\mathrm{\nabla }v),\hfill & x\in \mathrm{\Omega },t>0,\hfill \\ 0=\mathrm{\Delta }v-v+u,\hfill & x\in \mathrm{\Omega },t>0,\hfill \end{array}\right.\end{array}$$

under homogeneous Neumann boundary conditions in a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n,n\ge 2,$ with smooth boundary. Here, $\mathit{\delta }>0$ and the initial function $u(x,0)=u_0$ and the sensitivity function $\mathit{\chi }$ satisfy: $$\begin{array}{c}\hfill \begin{array}{cc}& u_0\in C^0(\overline{\mathrm{\Omega }})\text{with}{\int }_{\mathrm{\Omega }}{u}_0\mathrm{d}x>0,\hfill \\ \hfill & \mathit{\chi }(s)>0\text{for}s>0\text{and}\mathit{\chi }\in C^1([0,\infty )).\hfill \end{array}\end{array}$$

We prove that the classical solutions to the above system are uniformly in-time-bounded provided that there exists a smooth positive function $\mathit{\phi }$ such that for some $p>\frac{n}{2}$ and $0<\mathit{\lambda }<1,$ the following differential inequality holds $$\begin{array}{c}\hfill \mathit{\phi }(s)+(p-1)\mathit{\chi }(s)\le \sqrt{-\frac{4\mathit{\lambda }\mathit{\delta }(p-1)}{p}{\mathit{\phi }}^{\prime }(s)},s>0,\end{array}$$

where ${\mathit{\phi }}^{\prime }(s)<0$ and $s\mathit{\phi }(s)$ is bounded from above. We also present our results for the special case $0<\mathit{\chi }(s)\le \frac{{\mathit{\chi }}_0}{s^k}$ with ${\mathit{\chi }}_0>0$ and $k\ge 1.$ These results coincide with the results obtained by Fujie et al. [Math Methods Appl Sci. 2015] in the case of $k=1$ and extend their results in the case of $k>1$.

KEYWORDS:

• Chemotaxis
• singular sensitivity
• boundedness

AMS SUBJECT CLASSIFICATION:

• 35Kxx

Acknowledgements

The authors would like to thank the anonymous referees for their careful reading and valuable suggestions on this article.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The second author was supported by Institute for Research in Fundamental Sciences (IPM).