A quasilinear parabolic–elliptic chemotaxis-growth system with nonlinear secretion

ABSTRACT

This paper deals with the parabolic–elliptic chemotaxis-growth system with nonlinear secretion $\begin{array}{rl}{u}_t& =\mathrm{\nabla }\cdot \left(D\right(u\left)\mathrm{\nabla }u\right)-\mathrm{\nabla }\cdot \left(\chi u\mathrm{\nabla }v\right)+au-b{u}^r,x\in \mathrm{\Omega },t>0,\\ 0& =\mathrm{\Delta }v-v+u^k,x\in \mathrm{\Omega },t>0,\end{array}$ under homogeneous Neumann boundary conditions in a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n(n\ge 2)$ with smooth boundary, where $\chi >0$, a>0, b>0, r>1, $k\ge 1$ and $D\left(u\right)$ is a smooth function satisfying $D\left(u\right)>c_D{u}^{m-1}$ with some $c_D>0$ and $m\ge 1$. It is shown, either $r<k+1,m>k+1-\frac{2}{n},$ or $r>k+1,$ or $\begin{array}{rl}& r=k+1,b>b^{\ast },\mathrm{where}\\ & b^{\ast }=\left\{\begin{array}{ll}0,& \mathrm{if}m>k+1-\frac{2}{n},\\ \frac{n(k+1-m)-2}{n(k+1-m)},& \mathrm{if}m\le k+1-\frac{2}{n},\end{array}\right.\end{array}$ then the system admits a unique global bounded classical solution. Moreover, we obtain the large time behavior of the solution for a specific logistic source. Finally, for the case $D\left(u\right)\equiv 1$ and r=k+1, large time behavior and convergence rate are established.

KEYWORDS:

• Chemotaxis
• boundedness
• asymptotic behavior
• logistic source

AMS(2010) SUBJECT CLASSIFICATION:

• 92C17
• 35K55
• 35B40

Acknowledgments

The authors are very grateful to the anonymous reviewers for their careful reading and valuable comments which greatly improved this work.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by National Natural Science Foundation of China (NSFC) (Grant No. 11371384 and No. 11571062), the Basic and Advanced Research Project of CQC-STC (Grant No. cstc2015jcyjBX0007) and Fundamental Research Funds for the Central Universities (Grant Nos. 10611CDJXZ238826).