Global boundedness of solutions to a chemotaxis consumption model with signal dependent motility and logistic source - Baghaei, Khadijeh (Comptes Rendus. Mathématique)
This paper deals with the following chemotaxis system: \begin{equation*} {\left\lbrace \begin{array}{ll} u_{t}=\nabla \cdot \big (\gamma (v) \,\nabla u-u \,\xi (v) \,\nabla v\big )+\mu \, u\,(1-u), & x\in \Omega , \ t>0, \\ v_{t}=\Delta v-uv, & x\in \Omega , \ t>0, \end{array}\right.} \end{equation*} under homogeneous Neumann boundary conditions in a bounded domain $ \Omega \subset \mathbb{R}^{n}, n\ge 2,$ with smooth boundary. Here, the functions $\gamma (v)$ and $\xi (v)$ are as: \begin{equation*} \gamma (v)=(1+v)^{-k}\quad \mbox {and} \quad \xi (v)=-(1-\alpha )\,\gamma ^{\prime }(v), \end{equation*} where $k>0$ and $\alpha \in (0,1).$For the above system, we prove that the corresponding initial boundary value problem admits a unique global classical solution which is uniformly-in-time bounded. This result is obtained under some conditions on initial value $ v_{0}$ and $\mu $ and without any restriction on $k$ and $\alpha .$ The obtained result extends the recent results obtained for this problem.