Boundedness and stabilization in a quasilinear forager-exploiter model with volume-filling effects (Q2162147)

The authors study the forager-exploiter system with volume-filling effect \[ \begin{cases} u_t = \nabla \cdot ((u+1)^m \nabla u) - \nabla \cdot(u(1-u) \nabla w) , & \quad (x,t) \in \Omega \times (0,\infty),\\ v_t = \nabla \cdot ((v+1)^l \nabla v) - \nabla \cdot(v(1-v) \nabla u), & \quad (x,t) \in \Omega \times (0,\infty), \\ w_t = \Delta w - (u+v)w - \mu w + r(x,t), & \quad (x,t) \in \Omega \times (0,\infty), \end{cases} \] endowed with homogeneous Neumann boundary conditions and nonnegative initial data \(u_0, v_0, w_0 \in W^{2,\infty}(\Omega)\) satisfying the Neumann boundary conditions such that \(u_0 \not\equiv 0\), \(v_0 \not\equiv 0\), \(w_0 \not\equiv 0\), and \(u_0,v_0 \le 1\). Moreover, it is assumed that \(\Omega \subset \mathbb{R}^n\) is a bounded domain with smooth boundary with \(n \in \mathbb{N}\) as well as \(m,l \in \mathbb{R}\), \(\mu > 0\) and \(r \in C^1(\overline{\Omega} \times [0,\infty)) \cap L^\infty (\Omega \times (0,\infty))\). Here, \(u\) and \(v\) denote the densities of foragers and exploiters and \(w\) the nutrient concentration. The authors prove the existence of a unique global classical solution to this system such that all solution components are bounded in \(W^{1,\infty}(\Omega)\). If in addition \[ \int_t^{t+1} \int_\Omega r dx ds \to 0 \quad\text{as } t \to \infty \] is satisfied, then it is further shown that this global solution \((u,v,w) (t)\) converges to the constant steady state \((\frac{1}{|\Omega|} \int_\Omega u_0, \frac{1}{|\Omega|} \int_\Omega v_0, 0)\) in \((L^\infty (\Omega))^3\) as \(t \to \infty\). The proof of the global existence mainly relies on a combination of several a priori estimates for the solution components and their gradients.