Stability, bifurcation and spikes of stationary solutions in a chemotaxis system with singular sensitivity and logistic source
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| Natura: | Preprint |
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2024
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| _version_ | 1866914724683186176 |
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| author | Kurt, Halil Ibrahim Shen, Wenxian Xue, Shuwen |
| author_facet | Kurt, Halil Ibrahim Shen, Wenxian Xue, Shuwen |
| contents | 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}-χ(\frac{u}{v} v_x)_x+u(a-b u), & 0<x<L, \, t>0,\cr 0=v_{xx}- μv+ νu, & 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 $χ$, $a$, $b$, $μ$, $ν$ are positive constants. Among others, we prove there are $χ^*>0$ and $\{χ_k^*\}\subset [χ^*,\infty)$ ($χ^*\in\{χ_k^*\}$) such that the constant solution $(\frac{a}{b},\fracνμ\frac{a}{b})$ of (1) is locally stable when $0<χ<χ^*$ and is unstable when $χ>χ^*$, and under some generic condition, for each $k\ge 1$, a (local) branch of non-constant stationary solutions of (1) bifurcates from $(\frac{a}{b},\fracνμ\frac{a}{b})$ when $χ$ passes through $χ_k^*$, and global extension of the local bifurcation branch is obtained. We also prove that any sequence of non-constant positive stationary solutions $\{(u(\cdot;χ_n),v(\cdot;χ_n))\}$ of (1) with $χ=χ_n(\to \infty)$ develops spikes at any $x^*$ satisfying $\liminf_{n\to\infty} u(x^*;χ_n)>\frac{a}{b}$. Some numerical analysis is carried out. It is observed numerically that the local bifurcation branch bifurcating from $(\frac{a}{b},\fracνμ\frac{a}{b})$ when $χ$ passes through $χ^*$ can be extended to $χ=\infty$ and the stationary solutions on this global bifurcation extension are locally stable when $χ\gg 1$ and develop spikes as $χ\to\infty$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_14907 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stability, bifurcation and spikes of stationary solutions in a chemotaxis system with singular sensitivity and logistic source Kurt, Halil Ibrahim Shen, Wenxian Xue, Shuwen Analysis of PDEs Dynamical Systems 35B20, 35B32, 35B40, 35Q92, 92C17, 92D25 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}-χ(\frac{u}{v} v_x)_x+u(a-b u), & 0<x<L, \, t>0,\cr 0=v_{xx}- μv+ νu, & 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 $χ$, $a$, $b$, $μ$, $ν$ are positive constants. Among others, we prove there are $χ^*>0$ and $\{χ_k^*\}\subset [χ^*,\infty)$ ($χ^*\in\{χ_k^*\}$) such that the constant solution $(\frac{a}{b},\fracνμ\frac{a}{b})$ of (1) is locally stable when $0<χ<χ^*$ and is unstable when $χ>χ^*$, and under some generic condition, for each $k\ge 1$, a (local) branch of non-constant stationary solutions of (1) bifurcates from $(\frac{a}{b},\fracνμ\frac{a}{b})$ when $χ$ passes through $χ_k^*$, and global extension of the local bifurcation branch is obtained. We also prove that any sequence of non-constant positive stationary solutions $\{(u(\cdot;χ_n),v(\cdot;χ_n))\}$ of (1) with $χ=χ_n(\to \infty)$ develops spikes at any $x^*$ satisfying $\liminf_{n\to\infty} u(x^*;χ_n)>\frac{a}{b}$. Some numerical analysis is carried out. It is observed numerically that the local bifurcation branch bifurcating from $(\frac{a}{b},\fracνμ\frac{a}{b})$ when $χ$ passes through $χ^*$ can be extended to $χ=\infty$ and the stationary solutions on this global bifurcation extension are locally stable when $χ\gg 1$ and develop spikes as $χ\to\infty$. |
| title | Stability, bifurcation and spikes of stationary solutions in a chemotaxis system with singular sensitivity and logistic source |
| topic | Analysis of PDEs Dynamical Systems 35B20, 35B32, 35B40, 35Q92, 92C17, 92D25 |
| url | https://arxiv.org/abs/2403.14907 |