1. Global bounded solution of a forager–exploiter model with logistic sources and different taxis mechanisms.
- Author
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Liu, Changfeng and Guo, Shangjiang
- Subjects
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NEUMANN boundary conditions , *TAXICABS - Abstract
This paper deals with a forager–exploiter model with homogeneous Neumann boundary condition: u t = d 1 Δ u - ∇ · (χ (w) u ∇ w) + a 1 u - b 1 u θ 1 x ∈ Ω , t > 0 , v t = d 2 Δ v - ∇ · (ρ (u) v ∇ u) + a 2 v - b 2 v θ 2 x ∈ Ω , t > 0 , w t = d 3 Δ w - α (u + v) w - β w + r (x , t) x ∈ Ω , t > 0 , ∂ ν u = ∂ ν v = ∂ ν w = 0 x ∈ ∂ Ω , t > 0 , u (x , 0) = u 0 (x) ≥ 0 , v (x , 0) = v 0 (x) ≥ 0 w (x , 0) = w 0 (x) ≥ 0 x ∈ Ω , where Ω ⊂ R n (n ≥ 2) is a bounded domain, the constants a 1 , a 2 , b 1 , b 2 , θ 1 , θ 2 , α , β are positive. The nonnegative functions χ (w) , ρ (u) , r (x , t) satisfy χ (w) ∈ C 1 ([ 0 , ∞)) , ρ (u) ∈ C 1 ([ 0 , ∞)) and r (x , t) ∈ C 1 (Ω ¯ × [ 0 , ∞)) ∩ L ∞ (Ω × (0 , ∞)) , respectively. The nonnegative initial functions satisfy u 0 , w 0 ∈ W 2 , ∞ (Ω) , v 0 ∈ W 1 , ∞ (Ω) and ∂ ν u 0 = ∂ ν v 0 = ∂ ν w 0 = 0 . When n > 2 , θ 1 > 2 and θ 2 > n 2 + 1 , we prove that this problem possesses a global classical solution, which is uniformly bounded. This result improves the work of Wang (Nonlinear Anal 222: 112985, 2022), in which the global boundedness of solution is established for θ 1 > n 2 + 1 , θ 2 > n 2 + 1 , χ (w) ≡ χ and ρ (u) ≡ ρ and χ , ρ > 0 are two constants. Moreover, when n = 2 , either θ 1 = 2 and θ 2 > 2 or 1 < θ 1 < 2 , θ 2 > 2 and ∫ t t + 1 ∫ Ω | ∇ r (x , t) | 2 < ∞ for all t > 0 , this problem also possesses a global bounded classical solution. This result partially generalizes previously known ones. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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