Improvement of conditions for boundedness in a fully parabolic chemotaxis system with nonlinear signal production

Pan, Xu; Wang, Liangchen

Comptes Rendus. Mathématique, Tome 359 (2021) no. 2, pp. 161-168.

This paper deals with the chemotaxis system with nonlinear signal secretion

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

under homogeneous Neumann boundary conditions in a bounded domain $\Omega \subset {ℝ}^n$ ($n\ge 2$). The diffusion function $D\left(s\right)\in C^2\left([0,\infty )\right)$ and the chemotactic sensitivity function $S\left(s\right)\in C^2\left([0,\infty )\right)$ are given by $D\left(s\right)\ge C_d{(1+s)}^{-\alpha }$ and $0<S\left(s\right)\le C_{s}s{(1+s)}^{\beta -1}$ for all $s\ge 0$ with $C_d,C_s>0$ and $\alpha ,\beta \in ℝ$. The nonlinear signal secretion function $g\left(s\right)\in C^1\left([0,\infty )\right)$ is supposed to satisfy $g\left(s\right)\le C_g{s}^{\gamma }\text{for}\text{all}s\ge 0$ with $C_g,\gamma >0$. Global boundedness of solution is established under the specific conditions:

$$0<\gamma \le 1\text{and}\alpha +\beta <min\left\{1+\frac{1}{n},1+\frac{2}{n}-\gamma \right\}.$$

The purpose of this work is to remove the upper bound of the diffusion condition assumed in [9], and we also give the necessary constraint $\alpha +\beta <1+\frac{1}{n}$, which is ignored in [9, Theorem 1.1].

Classification : 35K35, 35A01, 35B44, 35B35, 92C17

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