Boundedness of classical solutions to a chemotaxis consumption system with signal dependent motility and logistic source

Baghaei, Khadijeh

Comptes Rendus. Mathématique, Tome 361 (2023) no. G10, pp. 1641-1652

We consider the chemotaxis system:

$$\left\{\begin{array}{cc}{u}_t=\nabla ·\left(\gamma \left(v\right)\nabla u-u\xi \left(v\right)\nabla v\right)+\mu u(1-u),\hfill & x\in \Omega ,t>0,\hfill \\ v_t=\Delta v-uv,\hfill & x\in \Omega ,t>0,\hfill \end{array}\right.$$

under homogeneous Neumann boundary conditions in a bounded domain $\Omega \subset {ℝ}^n,n\ge 2,$ with smooth boundary. Here, the functions $\gamma \left(v\right)$ and $\xi \left(v\right)$ are as:

$$\gamma \left(v\right)={(1+v)}^{-k}\text{and}\xi \left(v\right)=-(1-\alpha ){\gamma }^{\prime }\left(v\right),$$

where $k>0$ and $\alpha \in (0,1).$

We prove that the classical solutions to the above system are uniformly-in-time bounded provided that $k(1-\alpha )<\frac{4}{n+5}$ and the initial value $v_0$ and $\mu$ satisfy the following conditions:

$$\begin{array}{c}\hfill 0<\parallel v_0{\parallel }_{L^{\infty }\left(\Omega \right)}\le {\left[\frac{4\left[1-k\left(1-\alpha \right)\right]}{k(n+1)(1-\alpha )}\right]}^{\frac{1}{k}}-1,\end{array}$$

and

$$\mu >\frac{kn(1-\alpha )\parallel v_0{\parallel }_{L^{\infty }\left(\Omega \right)}}{(n+1)(1+\parallel v_0{\parallel }_{L^{\infty }\left(\Omega \right)})}.$$

This result improves the recent result obtained for this problem by Li and Lu (J. Math. Anal. Appl.) (2023).

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