Global boundedness and large time behavior of solutions to a chemotaxis–consumption system with signal-dependent motility

Abstract

This paper deals with the following chemotaxis–consumption system with signal-dependent motility $$\begin{aligned} \left\{ \begin{array}{llll} u_{t}=\Delta (\gamma (v)u),\,\,\, &{}x\in \Omega ,\,\,\, t>0,\\ v_{t}=\Delta v-uv, \,\,\,&{}x\in \Omega ,\,\,\, t>0,\\ \end{array} \right. \end{aligned}$$ under no-flux boundary conditions in a smoothly bounded domain $$\Omega \subset {\mathbb {R}}^{n}$$ . $$\gamma (s)$$ is the motility function. For the case of positive motility function, it is shown that the corresponding initial boundary value problem possesses a unique global classical solution which is uniformly bounded. Moreover, it is asserted that the solution to the system exponentially converges to constant equilibria in the large time. Finally, if the motility function is zero at some point, we obtain the existence of the weak solution.