关键词: spatial Solow-Swan model, density-dependent, global boundedness, large time behavior

Abstract: In this paper, we consider the following spatial Solow-Swan model with density-dependent motion $$\begin{align*} \left\{ \begin{aligned} &u_t=\Delta(\phi(v)u)+\mu u(1-u^\sigma), \quad &x&\in\Omega, t>0, \\ &v_t=\Delta v-v+u^{1-\alpha}v^\alpha, &x&\in\Omega, t>0, \\ &\frac{\partial u}{\partial \nu}=\frac{\partial v}{\partial \nu}=0, &x&\in\partial\Omega, t>0, \\ &u(x,0)=u_0(x), v(x,0)=v_0(x), &x&\in \Omega, \end{aligned} \right. \end{align*}$$ where $ \sigma>0, \alpha\in(0,1) $ and $ \Omega\subset \mathbb{R}^n $ $ (n\ge1) $ is a bounded domain with smooth boundary and $ \phi\in C^3([0,\infty)), \phi(s)>0 \text{ for all } s\ge0 $. We prove that if $$\begin{align*} \left\{ \begin{aligned} & \sigma+\alpha>\max\{1,\frac n2(1-\alpha)\}, &\text{ for }&\mu>0, \\ & 1-\alpha< \frac2n, &\text{ for }& \mu=0, \end{aligned} \right. \end{align*}$$ then there exists a unique time-globally classical solution $(u,v)$ for all $n\ge1$, such a solution is bounded and satisfies $u\ge0$, $v>0$. Moreover, we show that the above solution will convergence to the steady state $(1,1)$ exponentially in $L^\infty$ as $t\to\infty$.

Key words: spatial Solow-Swan model, density-dependent, global boundedness, large time behavior