Boundedness of classical solutions to a chemotaxis consumption model with signal-dependent motility

Abstract

This paper deals with the following chemotaxis system:

$$\begin{aligned} \left\{ \begin{array}{ll} u_{t}=\nabla \cdot \big (\gamma (v) \nabla u-u \,\xi (v) \nabla v\big )+\mu \, u\,(1-u), &{} x\in \Omega , \ t>0, \\ v_{t}=\Delta v-uv, &{} x\in \Omega , \ t>0, \end{array} \right. \end{aligned}$$

under homogeneous Neumann boundary conditions in a bounded domain \( \Omega \subset {\mathbb {R}}^{n}, n\ge 2,\) with smooth boundary. Here, the positive function \(\gamma \in C ^{2}([0, +\infty )) \) satisfies \(\gamma '(s)<0\) and \( \gamma ''(s)\ge 0\) for all \(s\ge 0,\) also \(\xi (s)= -(1-\alpha )\,\gamma '(s) \) with \(\alpha \in (0, 1)\). 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 for small initial data without any restriction on \(\mu .\) The obtained result improves a recent result by Li and Lu (J Math Anal Appl 521:126902, 2023), which asserts the global existence of bounded classical solutions, provided that \( \frac{(\gamma '(s))^{2}}{\gamma ''(s)} \le \frac{n}{2(n+1)^{3}}\) and some conditions on initial data and \(\mu .\) We should mention that in the special cases \(\gamma (s)=(1+s)^{-k}\,(k>0)\) and \(\gamma (s)=\textit{e}^{-\chi s}\, (\chi >0),\) the result in Li and Lu (2023) is obtained under conditions on k and \(\chi .\) But, our result is without any restriction on k and \(\chi .\)