2022/12/19 by Kurt, Halil Ibrahim, Shen, Wenxian
#Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2212.09838
This paper is concerned with 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. This is the first work on two-species chemotaxis-competition system with singular sensitivity and Lotka-Volterra competitive kinetics. Among others, we prove 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 (u(t,x;u0,v0),v(t,x;u0,v0),w(t,x;u0,v0)) with u(0,x;u0,v0)=u0(x) and v(0,x;u0,v0)=v0(x) provided that min\a1,a2\ is large relative to χ1,χ2 and u0+v0 is not small. Moreover, under the same condition, we prove that \limsupt→∞ ‖u(t,⋅;u0,v0)+v(t,⋅;u0,v0)‖_∞≤ M^*, and \liminft→∞ infx∈Ω(u(t,x,u0,v0)+v(t,x;u0,v0))≥ m^*, for some positive constants M^*,m^* independent of u0,v0, the latter is referred to as combined pointwise persistence.