2024/04/04 by Halil İbrahim Kurt, Wenxian Shen, Kurt, Halil Ibrahim +1
Mathematics · Biochemistry, Genetics and Molecular Biology · Engineering · #Mathematical Biology Tumor Growth #Gene Regulatory Network Analysis #Molecular Communication and Nanonetworks
paper · pdf · doi:10.48550/arxiv.2404.03158
The current paper is concerned with the stabilization in the following parabolic-parabolic-elliptic chemotaxis system with singular sensitivity and Lotka-Volterra competitive kinetics, \begincases ut=Δu-χ1 ∇⋅ ((u)/(w) ∇ w)+u(a1-b1u-c1v) , amp;x∈ Ω\cr vt=Δv-χ2 ∇⋅ ((v)/(w) ∇ w)+v(a2-b2v-c2u), amp;x∈ Ω\cr 0=Δw-μw +νu+ λv, amp;x∈ Ω\cr (∂ u)/(∂ n)=(∂ v)/(∂ n)=(∂ w)/(∂ n)=0, amp;x∈∂Ω, \endcases where Ω⊂ ℝN is a bounded smooth domain, and χi,ai, bi, ci (i=1,2) and μ, ν, λ are positive constants. In [25], we proved that for any given nonnegative initial data u0,v0∈ C0(Ω) with u0+v0\not ≡ 0, (0.1) has a unique globally defined classical solution provided that min\a1,a2\ is large relative to χ1,χ2, and u0+v0 is not small in the case that (χ1-χ2)2≤ max\4χ1,4χ2\ and u0+v0 is neither small nor big in the case that (χ1-χ2)2>max\4χ1,4χ2\. In this paper, we proved that (0.1) has a unique positive constant solution (u^*,v^*,w^*), where u^*=(a1b2-c1a2)/(b1b2-c1c2), v^*=(b1a2-a1c2)/(b1b2-c1c2), w^*=\fracνμu^*+\fracλμ v^*. We obtain some explicit conditions on χ1,χ2 which ensure that the positive constant solution (u^*,v^*,w^*) is globally stable in the sense that for any given nonnegative initial data u0,v0∈ C0(Ω) with u0\not ≡ 0 and v0\not ≡ 0, limt→∞(‖u(t,⋅;u0,v0)-u^*‖_∞ +‖v(t,⋅;u0,v0)-v^*‖_∞+‖w(t,⋅;u0,v0)-w^*‖_∞)=0.