Boundedness and asymptotic stability in a chemotaxis-haptotaxis system with flux limitation and nonlinear signal production

This paper deals with the chemotaxis-haptotaxis system with flux limitation and nonlinear signal production, $ \begin{align*} \begin{cases} u_t = \Delta u -\chi \nabla \cdot \left( \dfrac{u\nabla v}{\left(1 + |\nabla v |^2 \right)^{\alpha}} \right) -\xi \nabla \cdot (u\nabla w) + \mu u(1-u-w), \\ 0 = \Delta v - v + u^{\beta}, \\ w_t = -vw \end{cases} \end{align*} $ in a bounded domain $ \Omega \subset \mathbb{R}^n $ $ (n \geq 2) $ with smooth boundary, where $ \alpha \in \mathbb{R} $ and $ \chi, \xi, \mu, \beta>0 $ are given constants. The first result of this paper asserts global existence and boundedness of classical solutions $ (u, v, w) $ provided that either $ \beta n > 1, \quad \frac{1}{2} - \frac{1}{2(\beta n - 1)} < \alpha $ or $ \beta n \le 1. $ The second result shows that when $ \alpha \ge 0 $, under a smallness condition on the initial datum for $ w $, the function $ t \mapsto \|w(\cdot, t)\|_{W^{1, \infty}(\Omega)} $ decays exponentially as $ t \to \infty $.