Effects of density-suppressed motility in a two-dimensional chemotaxis model arising from tumor invasion
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
Effects of density-suppressed motility in a two-dimensional chemotaxis model arising from tumor invasion Dan Li and Chun Wu
Abstract. This paper deals with the global stability of the following density-suppressed motility system ⎧ ut = Δ(ϕ(v)u), ⎪ ⎪ ⎨ vt = Δv + wz, wt = −wz, ⎪ ⎪ ⎩ zt = Δz − z + u,
x ∈ Ω, x ∈ Ω, x ∈ Ω, x ∈ Ω,
t > 0, t > 0, t > 0, t>0
in a bounded domain Ω ⊂ R2 with smooth boundary, where the motility function ϕ(v) is positive. If ϕ(v) has the lowerupper bound, we can obtain that this system possesses a unique bounded classical solution. Moreover, we can obtain that the global solution (u, v, w, z) will exponentially converge to the equilibrium (u0 , v 0 + w0 , 0, u0 ) as t → +∞, where 1 f 0 = |Ω| Ω f0 (x)dx. AMS Subject Classification (2010). 35B35, 35B34, 35K55, 92C17. Keywords. Chemotaxis, Boundedness, Large time behavior, Tumor invasion.
1. Introduction In this paper, we will consider the initial-boundary value problem for density-dependent motility: ⎧ ut = ∇ · (ϕ(v)∇u) + ∇ · (χ(v)u∇v), x ∈ Ω, ⎪ ⎪ ⎪ ⎪ x ∈ Ω, vt = Δv + wz, ⎪ ⎪ ⎨ x ∈ Ω, wt = −wz, x ∈ Ω, zt = Δz − z + u, ⎪ ⎪ ⎪ ∂u ∂v ∂z ⎪ = = = 0, x ∈ ∂Ω, ⎪ ⎪ ∂ν ∂v ⎩ ∂ν (u, v, w, z)(x, 0) = (u0 , v0 , w0 , z0 )(x), x ∈ Ω,
the tumor invasion system with t > 0, t > 0, t > 0, t>0 t > 0,
(1.1)
∂ denotes the outward normal where Ω ⊂ R2 is a bounded domain with smooth boundary ∂Ω and ∂ν derivative on ∂Ω. System (1.1) was proposed by Fujie et al. [4] to describe a tumor invasion phenomenon with chemotaxis effect of Chaplain and Anderson type [1], where u(x, t), v(x, t), w(x, t) and z(x, t) denote the density of tumor cells, the concentration of active extracellular matrix (ECM∗ ), extracellular matrix (ECM) and matrix degrading enzymes (MDE), respectively. The u−equation of (1.1) describes the random motion of tumor cells with an ECM∗ -dependent motility coefficient ϕ(v).
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D. Li and C. Wu
ZAMP
Model (1.1) originates in, but essentially differs from the following density-suppressed motility system ⎧ ut = ∇ · (ϕ(v)∇u) + ∇ · (χ(v)u∇v), x ∈ Ω, t > 0, ⎪ ⎪ ⎨ x ∈ Ω, t > 0, vt = Δv − v + u, (1.2) ∂u ∂v = = 0, x ∈ ∂Ω, t > 0, ⎪ ⎪ ∂ν ∂ν ⎩ x ∈ Ω. (u, v)(x, 0) = (u0 , v0 ), If ϕ(v) ≡ 1 and χ(v) ≡ χ < 0 (see [5]), the solution of system (1.2) will exist globally or blow up in a infinite/finite time (no blow-up in 1-D [6], critical mass blow-up in 2-D [7–9,11] and blow-up in ≥3-D [22,23]). If ϕ(v) ≡ 1 and χ(v) = χv0 (χ0 > 0), Winkler [25] established global existence of classical solution for χ0 < n2 . Fujie [2] obtained the global boundedness of the solutions to (1.2); in [15], Lankeit improved these results in the two-dimensional setting. Some results such as global boundedness and asymptotic behavior of classical solutions are also obtained in [16,21,25]. For the case χ(v) = ϕ (v), minv∈[0,∞] ϕ(v) = 0 is possible under the biological assumption ϕ (v) < 0, and hence degeneracy may occur. This will hinder the understanding of global dynamics of solutions to (1.2).
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