On a quasilinear fully parabolic chemotaxis-type solow-swan model with indirect signal

In this paper, we consider the following fully parabolic chemotaxis-type Solow-Swan model with indirect signal$ \begin{eqnarray*} \left\{ \begin{aligned} &u_{t} = \Delta u-\chi \nabla \cdot (u^{\theta }\nabla v)+\mu_{1} u-\mu_{2}u^{\kappa },&(x,t)\in \Omega\times (0,\infty),\\ &v_{t} = \Delta v-v+\mu _{3}w^{1-\alpha }v^{\alpha },&(x,t)\in \Omega\times (0,\infty),\\ &w_{t} = \Delta w-w+u,&(x,t)\in \Omega\times (0,\infty), \end{aligned} \right. \end{eqnarray*} $under homogeneous Neumann boundary conditions in a smoothly bounded domain $ \Omega \subset \mathbb{R}^{n} $ $ (n\ge 1) $, where $ \chi>0 $, $ \mu_{1} $, $ \mu _{2}\ge0 $, $ \mu_{3}>0 $, $ \theta\ge1,\kappa>2 $, and $ \alpha\in(0,1) $.$ \bullet $ When $ \mu_{1} = \mu_{2} = 0 $, we prove that this problem possesses a global classical solution that is uniformly bounded in time under the conditions of $ n = 2 $, $ \alpha\in(0,1) $, $ \theta = 1 $ or $ n>2 $, $ 1-\alpha\le\frac{1}{n} $, $ \theta<1+\frac{1}{n} $.$ \bullet $ When $ \mu_{1},\mu_{2}>0 $, we also demonstrate that this problem possesses a globally bounded classical solution if $ \theta\le\kappa-1 $.