Global classical solution and stability of a forager-exploiter model with double-taxis effects

This paper is concerned with the global solvability and stability of classical solutions for the following forager-exploiter model with double-taxis effects,

$ \begin{align*} \left\{ \begin{aligned} &u_t = \Delta u-\chi\nabla\cdot(u\nabla w), \\ &v_t = \Delta v -\xi\nabla\cdot(v\nabla u)+\mu (v-v^{2}), \\ &0 = \Delta w-(u+v)w-\theta w+r(x, t). \end{aligned}\right. \end{align*} $

The current conclusions on global solvability primarily rely on the inclusion of specific higher-order damping terms, such as $ \eta(u - u^m) $ and $ \mu(v - v^l) $ with $ m, l>2 $, within the framework of fully parabolic structures. However, proving the global existence of classical solutions for the case where $ m = l = 2 $, or in the absence of logistic sources, remains a significant challenge. In this paper, we focus on the case where the third equation exhibits an elliptic structure. For this, we establish the global existence of classical solutions in the two-dimensional case. For the higher-dimensional cases ($ N\ge3 $), when $ \frac{\xi^2}{\mu} $ is sufficiently small, the classical solutions also exist globally. Moreover, we prove that the classical solutions converge exponentially to a constant equilibrium state when $ \frac{\chi}{\mu} $, $ \xi $ are sufficiently small and $ r(x, t) $ is a positive constant.