On the existence of globally bounded solutions for a spatial Solow-Swan model with density-dependent motion

This research examines a spatial Solow-Swan model characterized by density-dependent motion, as illustrated by the following system $\left(\begin{array}{cccc}& u_t=\nabla \cdot (\gamma \left(v\right)\nabla u-u\varphi \left(v\right)\nabla v)+\mu u(1-u^{\sigma }),& & x\in \Omega ,t>0,\\ & v_t=\Delta v-v+u^{1-\alpha }{v}^{\alpha },& & x\in \Omega ,t>0,\\ \end{array}\right)$where $\sigma >0,\alpha \in (0,1)$ are positive constants, $\Omega \subset R^N(N\ge 1)$ denotes a bounded domain with a smooth boundary, and the functions $\varphi$ and $\gamma$ belong to $C^3\left([0,\infty )\right)$ with $\varphi \left(s\right),\gamma \left(s\right)>0$ for all $s\ge 0$. The conditions for the parameters are specified as follows: $\left(\begin{array}{cccc}& \sigma +\alpha >\frac{N}{2}(1-\alpha ),& & \text{for}\mu >0,\\ & 1-\alpha <\frac{2}{N},& & \text{for}\mu =0.\end{array}\right)$Under Neumann boundary conditions, a unique globally bounded classical solution is established. Additionally, we illustrate that the solution to the aforementioned system converges exponentially to the steady state $(1,1)$ in $L^{\infty }\left(\Omega \right)$ as $t\to \infty$.