Abstract
under homogeneous Neumann boundary conditions in a bounded domain Ω⊂ℝ n (n≥2). The diffusion function D(s)∈C 2 ([0,∞)) and the chemotactic sensitivity function S(s)∈C 2 ([0,∞)) are given by D(s)≥C d (1+s) -α and 0<S(s)≤C s s(1+s) β-1 for all s≥0 with C d ,C s >0 and α,β∈ℝ. The nonlinear signal secretion function g(s)∈C 1 ([0,∞)) is supposed to satisfy g(s)≤C g s γ foralls≥0 with C g ,γ>0. Global boundedness of solution is established under the specific conditions:
Highlights
In the present work, we consider the following system, which describes the fully parabolic chemotaxis system with nonlinear diffusion, sensitivity and signal secretion ut = ∇ · (D(u)∇u − S(u)∇v), x ∈ Ω, t > 0, vt = ∆v − v + g (u), ∂u ∂ν = ∂v ∂ν= 0, x ∈ Ω, t > 0, x ∈ ∂Ω, t > 0, (1)(u, v)(x, 0) = (u0(x), v0(x)), x ∈ Ω, with homogeneous Neumann boundary conditions, where Ω ⊂ Rn (n ≥ 2) is a bounded domain, and ∂/∂ν is the derivative of the normal with respect to ∂Ω
This paper deals with the chemotaxis system with nonlinear signal secretion ut = ∇ · (D(u)∇u − S(u)∇v), x ∈ Ω, t > 0, vt = ∆v − v + g (u), x ∈ Ω, t > 0, under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ Rn (n ≥ 2)
The nonlinear signal secretion function g (s) ∈ C 1([0, ∞)) is supposed to satisfy g (s) ≤ Cg sγ for all s ≥ 0 with Cg, γ > 0
Summary
This paper deals with the chemotaxis system with nonlinear signal secretion ut = ∇ · (D(u)∇u − S(u)∇v), x ∈ Ω, t > 0, vt = ∆v − v + g (u), x ∈ Ω, t > 0, under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ Rn (n ≥ 2). Global boundedness of solution is established under the specific conditions: 0 < γ ≤ 1 and α + β < min 1 + 1 , 1 + 2 − γ . The purpose of this work is to remove the upper bound of the diffusion condition assumed in [9], and we give the necessary constraint α
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