2024/03/22 by Kurt, Halil Ibrahim, Shen, Wenxian, Xue, Shuwen · 1 citation
#35B20 #35B32 #35B40 #35Q92 #92C17 #92D25 #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2403.14907
In the current paper, we study stability, bifurcation, and spikes of positive stationary solutions of the following parabolic-elliptic chemotaxis system with singular sensitivity and logistic source, \begincases ut=uxx-χ((u)/(v) vx)x+u(a-b u), & 00,\cr 0=vxx- μv+ νu, & 00 \cr ux(t,0)=ux(t,L)=vx(t,0)=vx(t,L)=0, & t>0, \endcases where χ, a, b, μ, ν are positive constants. Among others, we prove there are χ^*>0 and \χk^*\⊂ [χ^*,∞) (χ^*∈\χk^*\) such that the constant solution ((a)/(b),\fracνμ(a)/(b)) of (1) is locally stable when 0<χ<χ^* and is unstable when χ>χ^*, and under some generic condition, for each k≥ 1, a (local) branch of non-constant stationary solutions of (1) bifurcates from ((a)/(b),\fracνμ(a)/(b)) when χ passes through χk^*, and global extension of the local bifurcation branch is obtained. We also prove that any sequence of non-constant positive stationary solutions \(u(⋅;χn),v(⋅;χn))\ of (1) with χ=χn(→ ∞) develops spikes at any x^* satisfying \liminfn→∞ u(x^*;χn)>(a)/(b). Some numerical analysis is carried out. It is observed numerically that the local bifurcation branch bifurcating from ((a)/(b),\fracνμ(a)/(b)) when χ passes through χ^* can be extended to χ=∞ and the stationary solutions on this global bifurcation extension are locally stable when χ≫ 1 and develop spikes as χ→∞.