AI 中文总结
本文针对带弱奇异敏感性与逻辑斯蒂源的趋化系统，通过变换z=v^(1−α)，证明其解在特定条件下收敛，且部分情况下收敛具指数性。
AI 中文摘要
本文研究光滑有界凸区域Ω⊂ℝⁿ（n≥2）上带齐次Neumann边界的抛物-椭圆型趋化系统：uₜ=Δu−χ∇⋅(u/v^α ∇v)+ru−μu²，0=Δv−v+u，其中α∈(0,1)，χ、r、μ>0。若α∈(0,(n+2)/(2n))且μ>足够大的μ₀，在未建立v的一致正下界的情况下，给出u关于系数μ的显式上界；通过变换z=v^(1−α)处理对应非奇异趋化系统，证明当α∈(0,1/2)且μ>足够大的μ_⋆时，解(u,v)在L^∞范数下随t→∞收敛到(r/μ, r/μ)，且当α∈(0,(n+2)/(n²+4))时该收敛具指数性。
英文摘要
This paper is concerned with the parabolic-elliptic chemotaxis system with weakly singular sensitivity and logistic source:~$ u_t=Δu-χ\nabla\cdot(\frac{u}{v^α}\nabla v) +ru-μu^2$, $0=Δv-v+u,$ under the homogeneous Neumann boundary in a smooth bounded convex domain $Ω\subset\mathbb{R}^n$ for $n\ge 2$. where $α\in(0,1)$ and $χ,r,μ>0$. If $α\in(0,\frac{n+2}{2n})$ and $μ>μ_0$ with $μ_0>0$ suitably large, we give the explicit expression of the upper bound for $u$ with respect to the coefficient $μ$ after some time, without establishing the uniformly positive bound for $v$ from below. Furthermore, by dealing with the corresponding non-singular chemotaxis system via the transformation $z=v^{1-α}$, it is proved that the solution $(u,v)$ converges to $(\frac{r}μ,\frac{r}μ)$ in $L^\infty$-norm as $t\rightarrow\infty$ if $α\in(0,\frac{1}{2})$ and $μ>μ_\star$ sufficiently large, which is moreover enjoying exponential convergence when $α\in(0,\frac{n+2}{n^2+4})$.
