Boundedness and stabilization enforced by mild saturation of taxis in a producer–scrounger model. (February 2021) - Cao, Xinru; Tao, Youshan
Abstract: With successive coupling of two taxis mechanisms and with density-dependent taxis sensitivities, the producer–scrounger system u t = Δ u − ∇ ⋅ ( u ( u + 1 ) β − 1 ∇ w ), v t = Δ v − ∇ ⋅ ( v ( v + 1 ) α − 1 ∇ u ), w t = Δ w − ( u + v ) w − λ w + g ( x, t ), is considered in a bounded smooth domain Ω ⊂ R n ( n ≥ 2 ), where α ≤ 1, β ≤ 1 and λ > 0 are prescribed parameters, and g is a given C 1 -smooth function. It is shown that whenever β < 1 and α < n + 2 2 n and g satisfies ∫ t t + 1 ∫ Ω g 2 → 0 as t → ∞, for all suitably regular initial data ( u 0, v 0, w 0 ) the corresponding homogeneous Neumann initial–boundary problem admits a global classical solution ( u, v, w ) that converges to ( 1 | Ω | ∫ Ω u 0, 1 | Ω | ∫ Ω v 0, 0 ) in the large time limit.