Boundedness in a chemotaxis system with weak singular sensitivity and logistic kinetics in any dimensional setting

This paper deals with the following parabolic-elliptic chemotaxis competition system with weak singular sensitivity and logistic source(0.1)$\{\begin{array}{cc}{u}_t=\Delta u-\chi \nabla \cdot \left(\frac{u}{v^{\lambda }}\nabla v\right)+ru-\mu u^2,& x\in \Omega ,\\ 0=\Delta v-\alpha v+\beta u,& x\in \Omega ,\\ \frac{\partial u}{\partial \nu }=\frac{\partial v}{\partial \nu }=0,& x\in \partial \Omega ,\end{array}$ where $\Omega \subset R^N(N\ge 1)$ is a smooth bounded domain, the parameters $\chi ,r,\mu ,\alpha ,\beta$ are positive constants and $\lambda \in (0,1)$.

It is well known that for parabolic-elliptic chemotaxis systems including singularity, a uniform-in-time positive pointwise lower bound for v is vitally important for establishing the global boundedness of classical solutions since the cross-diffusive term becomes unbounded near $v=0$. To this end, a key step in the literature is to establish a proper positive lower bound for the mass functional ${\int }_{\Omega }u$, which, due to the presence of logistic kinetics, is not preserved and hence it turns in for v. In contrast to this approach, in this article, the boundedness of classical solutions of (0.1) is obtained without using the uniformly positive lower bound of v.

Among others, it has been proven that without establishing a uniform-in-time positive pointwise lower bound for v, if $\lambda \in (0,1)$, then there exists $\mu >{\mu }^{⁎}$ such that for all suitably smooth initial data, $L^p$-norm (for any $p\ge 2$) of any globally defined positive solution is bounded; moreover, problem (0.1) possesses a unique globally defined classical solution. In addition, the solutions are shown to be uniformly bounded under the additional assumption$\lambda <\frac{1}{2}+\frac{1}{N}\text{with}N\ge 2.$