On boundedness, blow-up and convergence in a two-species and two-stimuli chemotaxis system with/without loop
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Calculus of Variations
On boundedness, blow-up and convergence in a two-species and two-stimuli chemotaxis system with/without loop Ke Lin1 · Tian Xiang2 Received: 12 September 2019 / Accepted: 30 April 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this work, we study dynamic properties of classical solutions to a homogenous Neumann initial-boundary value problem (IBVP) for a two-species and two-stimuli chemotaxis model with/without chemical signalling loop in a 2D bounded and smooth domain. We detect the product of two species masses as a feature to determine boundedness, gradient estimate, blow-up and exponential convergence of classical solutions for the corresponding IBVP. More specifically, we first show generally a smallness on the product of both species masses, thus allowing one species mass to be suitably large, is sufficient to guarantee global boundedness, higher order gradient estimates and W j,∞ ( j ≥ 1)-exponential convergence with rates of convergence to constant equilibria; and then, in a special case, we detect a straight line of masses on which blow-up occurs for large product of masses. Our findings provide new understandings about the underlying model, and thus, improve and extend greatly the existing knowledge relevant to this model. Mathematics Subject Classification Primary 35K59 · 35B25 · 35B44 · 35K51; Secondary 92C17 · 92D25
1 Introduction and statement of main results In this work, we further study dynamic properties of classical solutions to the Neumann initial-boundary value problem for the following two-species and two-stimuli chemotaxis
Communicated by P. Rabinowitz.
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Tian Xiang [email protected] Ke Lin [email protected]
1
School of Economics and Mathematics, Southwestern University of Economics and Finance, Chengdu 610074, Sichu, China
2
Institute for Mathematical Sciences, Renmin University of China, Bejing 100872, China 0123456789().: V,-vol
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model with/without chemical signalling loop: ⎧ ⎪ ⎪u t = ∇ · (∇u − χ1 u∇v) ⎪ ⎪ ⎪ τ1 vt = v − v + w ⎪ ⎪ ⎪ ⎨w = ∇ · (∇w − χ w∇z − χ w∇v) t 2 3 ⎪τ2 z t = z − z + u ⎪ ⎪ ⎪ ∂u ∂v ∂z ⎪ ⎪ = ∂ν = ∂w ⎪ ∂ν = ∂ν = 0 ⎪ ⎩ ∂ν (u, τ1 v, w, τ2 z) = (u 0 , τ1 v0 , w0 , τ2 z 0 )
in × (0, ∞), in × (0, ∞), in × (0, ∞), in × (0, ∞), on ∂ × (0, ∞), in × {0}.
(1.1)
∂ Here, ⊂ R2 is a bounded and smooth domain and ∂ν denotes the outer normal derivative on the boundary ∂, u = u(x, t) and w = w(x, t) respectively denote the unknown density of macrophages and tumor cells, while v = v(x, t) and z = z(x, t) represent the concentration of chemical signals secreted by w and u, respectively. The modelling parameters χi > 0, τi ≥ 0 (i = 1, 2) and χ3 ∈ R are given constants. Model (1.1) involves four unknown variables u, v, w, z and describes a two-species and two-stimuli chemotaxis model with/without chemical signalling loop, depending on χ3 = 0 or not: macrophages u secrete a chemical signal z, called gradient epidermal growth factor, which has an attractive impact on tumor
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