Global boundedness for an indirect consumption chemotaxis model with signal-dependent motility

In this paper, we consider the following chemotaxis system with signal-dependent motility and indirect signal consumption$\{\begin{array}{cc}{u}_t=\Delta \left(D\right(u\left)\varphi \right(v\left)\right),& (x,t)\in \Omega \times (0,\infty ),\\ v_t=\Delta v-vw,& (x,t)\in \Omega \times (0,\infty ),\\ w_t=\Delta w-w+u,& (x,t)\in \Omega \times (0,\infty ),\end{array}$ under the smooth bounded domain $\Omega \subset R^n(n\le 3)$ with homogeneous Neumann boundary conditions, where the nonlinearities $D\left(u\right)$ and the motility function ϕ satisfy the following condition$D\left(u\right)={(u+a)}^m\text{and}\varphi \in C^1\left(\right[0,\infty \left)\right)\text{is positive on}[0,\infty ).$ It has been demonstrated that, for any sufficiently regular initial data, the associated initial-boundary value problem allows for global classical solutions. Moreover, the asymptotic behavior of the solutions is analyzed and studied.