Boundedness and stabilization in a two‐species chemotaxis‐competition system of parabolic‐parabolic–elliptic type
Boundedness and stabilization in a two‐species chemotaxis‐competition system of parabolic‐parabolic–elliptic type
复制标题
DOI：
10.1002/mma.4607
复制
发表时间：
2017-03
影响因子：
2.9
通讯作者：
M. Mizukami
中科院分区：
文献类型：
--
作者：
M. Mizukami
研究两种群趋化竞争系统ut=d1Δu−χ1·（u w）+μ1u（1−u− a1 v）inΩ×（0，∞），vt=d2Δv−χ2·（v w）+μ 2 v（1− a2 u −v）inΩ×（0，∞），0 =d3Δw+αu+βv−γwinΩ×（0，∞），其中Ω是Rn中的有界区域，光滑边界<$Ω，n≥2; χi和μi是满足一定条件的常数.在a1，a2∈（0，1）且a1>1>a2的情形下，研究了上述系统，证明了当χiμi较小时，系统的全局存在性和渐近稳定性.然而，上述两种情况下的条件强烈地依赖于a1，a2，并且在a1，a2≥1的情况下还没有得到。此外，对于a1，a2∈（0，1）和a1>1>a2的收敛速度也没有研究。本文的目的是构造条件，使得对于所有a1，a2>0，a1，a2≥1的情形，经典有界解的整体存在性，并在a1，a2∈（0，1）和a1≥1>a2的情形下，得到上述方程组解的收敛速度.
This paper is concerned with the two‐species chemotaxis‐competition system ut=d1Δu−χ1∇·(u∇w)+μ1u(1−u−a1v)inΩ×(0,∞),vt=d2Δv−χ2∇·(v∇w)+μ2v(1−a2u−v)inΩ×(0,∞),0=d3Δw+αu+βv−γwinΩ×(0,∞), where Ω is a bounded domain in Rn with smooth boundary ∂Ω, n≥2; χi and μi are constants satisfying some conditions. The above system was studied in the cases that a1,a2∈(0,1) and a1>1>a2, and it was proved that global existence and asymptotic stability hold when χiμi are small. However, the conditions in the above 2 cases strongly depend on a1,a2, and have not been obtained in the case that a1,a2≥1. Moreover, convergence rates in the cases that a1,a2∈(0,1) and a1>1>a2 have not been studied. The purpose of this work is to construct conditions which derive global existence of classical bounded solutions for all a1,a2>0 which covers the case that a1,a2≥1, and lead to convergence rates for solutions of the above system in the cases that a1,a2∈(0,1) and a1≥1>a2.