Boundedness of the Higher-Dimensional Quasilinear Chemotaxis System with Generalized Logistic Source
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
BOUNDEDNESS OF THE HIGHER-DIMENSIONAL QUASILINEAR CHEMOTAXIS SYSTEM WITH GENERALIZED LOGISTIC SOURCE∗
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Qiao XIN (
Qingquan TANG (
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College of Mathmatics and Statistics, Yili Normal University, Yining 835000, China E-mail : [email protected]
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Chunlai MU (
College of Mathmatics and Statistics, Chongqing University, Chongqing 401331, China E-mail : [email protected] Abstract This article considers the following higher-dimensional quasilinear parabolicparabolic-ODE chemotaxis system with generalized Logistic source and homogeneous Neumann boundary conditions u = ∇ · (D(u)∇u) − ∇ · (S(u)∇v) + f (u), x ∈ Ω, t > 0 t vt = ∆v + w − v, x ∈ Ω, t > 0, wt = u − w, x ∈ Ω, t > 0, in a bounded domain Ω ⊂ Rn (n ≥ 2) with smooth boundary ∂Ω, where the diffusion coefficient D(u) and the chemotactic sensitivity function S(u) are supposed to satisfy D(u) ≥ M1 (u + 1)−α and S(u) ≤ M2 (u + 1)β , respectively, where M1 , M2 > 0 and α, β ∈ R. Moreover, the logistic source f (u) is supposed to satisfy f (u) ≤ a − µuγ with µ > 0, γ ≥ 1, and a ≥ 0. As α + 2β < γ − 1 + 2γ , we show that the solution of the above chemotaxis system n with sufficiently smooth nonnegative initial data is uniformly bounded. Key words
Chemotaxis system; logistic source; global solution; boundedness
2010 MR Subject Classification
1
35A01; 35K55; 35Q92
Introduction
In this article, we consider the following quasilinear parabolic-parabolic-ODE chemotaxis system with generalized logistic source ut = ∇ · (D(u)∇u) − ∇ · (S(u)∇v) + f (u), x ∈ Ω, t > 0, x ∈ Ω, t > 0, vt = ∆v + w − v, wt = u − w, x ∈ Ω, t > 0, (1.1) ∂u ∂v ∂w = = = 0, x ∈ ∂Ω, t > 0, ∂ν ∂ν ∂ν u(x, 0) = u (x), v(x, 0) = v (x), w(x, 0) = w (x), x ∈ Ω, 0 0 0 ∗ Received
November 27, 2018; revised May 15, 2019. This work is supported by the Youth Doctor Science and Technology Talent Training Project of Xinjiang Uygur Autonomous Region (2017Q087). † Corresponding author
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ACTA MATHEMATICA SCIENTIA
Vol.40 Ser.B
∂ denotes the derivative in a bounded domain Ω ⊂ Rn (n ≥ 2) with smooth boundary ∂Ω. ∂ν with respect to the outer normal of ∂Ω. The diffusion coefficient D(u) and the chemotactic sensitivity S(u) are satisfying that
D(u) ≥ M1 (u + 1)−α , for all u ≥ 0
(1.2)
S(u) ≤ M2 (u + 1)β , for all u ≥ 0
(1.3)
with M1 > 0 and α ∈ R,
with M2 > 0 and β ∈ R, as well as the logistic source f (u) is smooth satisfying f (0) ≥ 0 and f (u) ≤ a − µuγ , for all u ≥ 0
(1.4)
with a ≥ 0, µ > 0, and γ ≥ 1. For the nonnegative initial data, we assume that u0 ∈ C 0 (Ω), v0 ∈ W 1,∞ (Ω) and w0 ∈ W 1,∞ (Ω).
(1.5)
The original model of chemotaxis system (1.1) was proposed by Strohm, Tyson, and Powell [1] to describe the aggregation and spread behavior of the Mountain Pine Beetle (MPB), u(x, t) denotes the density of the flying MPB, v(x, t) stands for the concentration of the beetle pheromone, and w(x, t) represents the density of the nesting M
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