Stability, bifurcation and spikes of stationary solutions in a chemotaxis system with singular sensitivity and logistic source

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, \begin{cases} u_t=u_{xx}-\chi (\frac{u}{v} v_x)_x+u(a-b u), & 0<x<L, \, t>0,\cr 0=v_{xx}- \mu v+ \nu u, & 0<x<L, \, t>0 \cr u_x(t,0)=u_x(t,L)=v_x(t,0)=v_x(t,L)=0, & t>0, \tag{1} \end{cases} where $\chi$, $a$, $b$, $\mu$, $\nu$ are positive constants. Among others, we prove there are $\chi^*>0$ and $\{\chi_k^*\}\subset [\chi^*,\infty)$ ($\chi^*\in\{\chi_k^*\}$) such that the constant solution $(\frac{a}{b},\frac{\nu}{\mu}\frac{a}{b})$ of (1) is locally stable when $0<\chi<\chi^*$ and is unstable when $\chi>\chi^*$, and under some generic condition, for each $k\ge 1$, a (local) branch of non-constant stationary solutions of (1) bifurcates from $(\frac{a}{b},\frac{\nu}{\mu}\frac{a}{b})$ when $\chi$ passes through $\chi_k^*$, and global extension of the local bifurcation branch is obtained. We also prove that any sequence of non-constant positive stationary solutions $\{(u(\cdot;\chi_n),v(\cdot;\chi_n))\}$ of (1) with $\chi=\chi_n(\to \infty)$ develops spikes at any $x^*$ satisfying $\liminf_{n\to\infty} u(x^*;\chi_n)>\frac{a}{b}$. Some numerical analysis is carried out. It is observed numerically that the local bifurcation branch bifurcating from $(\frac{a}{b},\frac{\nu}{\mu}\frac{a}{b})$ when $\chi$ passes through $\chi^*$ can be extended to $\chi=\infty$ and the stationary solutions on this global bifurcation extension are locally stable when $\chi\gg 1$ and develop spikes as $\chi\to\infty$.