Abstract:

In this paper, we consider the spatial Solow system

$ \begin{cases} u_t = \Delta u - \chi \nabla \cdot (u \nabla v) + \mu u(1 - u^\sigma), & x \in \Omega, \; t > 0, \\ v_t = \Delta v - v + k u^{1-\alpha} v^\alpha, & x \in \Omega, \; t > 0 \end{cases} $

under homogeneous Neumann boundary conditions in a bounded domain $\Omega \subset \mathbb{R}^{N}$ with $N \geq 1$, where $\chi > 0$, $\mu > 0$, $k > 0$, $\alpha \in (0, 1)$, and $\sigma > 0$. We first establish that for $N \leq 2$, the corresponding Neumann initial-boundary value problem admits a global bounded classical solution $(u, v)$ with initial data $(u, v)|_{t=0} = (u_0, v_0)$ for all sufficiently regular initial data. Furthermore, for $N \geq 3$, under the additional hypotheses that

$ \mu > 0 \quad \text{and} \quad \max\left\{\sigma, \frac{2}{N}\right\} > \frac{(1 - \alpha)N}{N - \alpha(N - 2)}, $

we demonstrate that the aforementioned problem also possesses a unique global bounded classical solution.

Key words: Spatial Solow-Swan model, Chemotaxis system, Global existence, Boundedness