Global existence and boundedness of a chemotaxis model with indirect production and general kinetic function
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
Global existence and boundedness of a chemotaxis model with indirect production and general kinetic function Xie Li Abstract. This paper is devoted to the chemotaxis model with indirect production and general kinetic function ⎧ u = Δu − χ∇ · (u∇v) + f (u), ⎪ ⎨ t vt = Δv − v + w, ⎪ ⎩ τ wt + λw = g(u),
x ∈ Ω, t > 0, x ∈ Ω, t > 0, x ∈ Ω, t > 0,
in a bounded domain Ω ⊂ Rn (n ≤ 3) with smooth boundary ∂Ω, where χ, τ, λ are given positive parameters, f and g are known functions. We find several explicit conditions involving the kinetic function f , g, the parameters χ, λ, and the initial data u0 L1 (Ω) to ensure the global-in-time existence and uniform boundedness for the corresponding 2D/3D Neumann initial-boundary value problem. Particularly, when f ≡ 0, and g is a linear function, the global bounded classical solutions to the corresponding 2D Neumann initial-boundary value problem with arbitrarily large initial data and chemotactic sensitivity are established. Our results partially extend the results of Hu and Tao (Math Models Methods Appl Sci 26:2111–2128, 2016), Tao and Winkler (J Eur Math Soc 19:3641–3678, 2017), etc. Mathematics Subject Classification. 35A01, 35B40, 35B45, 35Q92, 92C17. Keywords. Chemotaxis, Indirect signal production, Kinetic function, Global existence, Boundedness.
1. Introduction and main results Motivated by the model for mountain pine beetle dispersal proposed by Strohm–Tyson–Powell (see Ref.[1]), the present work devotes to the chemotaxis model with indirect signal production and general kinetic function ⎧ ut = Δu − χ∇ · (u∇v) + f (u), x ∈ Ω, t > 0, ⎪ ⎪ ⎪ ⎪ ⎪ x ∈ Ω, t > 0, ⎪ ⎨ vt = Δv − v + w, τ wt = −λw + g(u), ⎪ ⎪ ⎪ ∂ν u = ∂ν v = 0, ⎪ ⎪ ⎪ ⎩ u(x, 0) = u0 (x), v(x, 0) = v0 (x), τ w(x, 0) = τ w0 (x),
x ∈ Ω, t > 0,
(1.1)
x ∈ ∂Ω, t > 0, x ∈ Ω,
n
in a bounded domain Ω ⊂ R (n ≤ 3) with smooth boundary ∂Ω, where χ, λ and τ are given positive parameters, the kinetic functions f and g belong to C 1 [0, +∞) satisfying f (0) ≥ 0, g(s) ≥ 0 for all s ≥ 0. The initial data u0 (x), v0 (x), and w0 (x) are given nonnegative functions satisfying: u0 (x) ∈ C(Ω), v0 (x) ∈ C 1 (Ω), w0 (x) ∈ C 1 (Ω), u0 ≥ 0, v0 ≥ 0, w0 ≥ 0 and u0 ≡ 0.
(1.2)
System (1.1) describes the spatio-temporal evolution of the mountain pine beetle (MPB) in forest habitat, where u represents the density of flying MPB, w denotes the density of nesting MPB which transited from the flying MPB with transition rate g, and v stands for the concentration of beetle 0123456789().: V,-vol
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pheromone secreted by nesting MPB, which attracts flying MPB to aggregate. The flying MPB undergo birth and death, and some of them may convert into nesting MPB as aforesaid, all these kinetic behaviors are described by the kinetic function f . Different from the well-known Keller–Segel model, the chemotactic cue, in such model, is not directly produced by flying MPB themselves, but indirectly secreted by nesting MPB. Actually, such indirect signaling mechanism also occu
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