Boundedness in a Forager-Exploiter Model Accounting for Gradient-Dependent Flux-Limitation

Abstract

The forager-exploiter model with gradient-dependent flux-limitation $$u_t=\Delta u-\chi\nabla \cdot (uk_f(1+|\nabla w|^2)^{-\frac{\alpha}{2}})\nabla w,$$$$v_t=\Delta v-\xi\nabla \cdot (vk_g(1+|\nabla u|^2)^{-\frac{\beta}{2}})\nabla u,$$$$w_t=\Delta w-(u+v)w-\mu w+r(x,t)$$is considered in smooth bounded domains $Ω ⊂ \mathbb{R}^N,$ $N ≥ 2.$ It is shown that if $α > (N − 2)/N(N − 1),$ $β > 0,$ then for any nonnegative functions $u_0,$ $v_0,$ $w_0∈ W^{2,∞}(Ω)$ such that $u_0 \not\equiv 0$ and $v_0 \not\equiv 0,$ the problem has a global classical solution $(u, v, w) ∈ (C^0 (\overline{Ω} × [0,∞))\cap C^{2,1}(\overline{Ω} × (0,∞)))^3$.