arXiv 1503.023872015-03-09math.AP

具Logistic源的高维拟线性全抛物型Keller-Segel系统的有界性

Boundedness in a quasilinear fully parabolic Keller-Segel system of higher dimension with logistic source

Cibing Yang, Xinru Cao, Zhaoxin Jiang, Sining Zheng

摘要：

本文研究了在非负初值和齐次Neumann边界条件下，含有Logistic型源项$ u_t=\nabla\cdot(ϕ(u)\nabla u)-χ\nabla\cdot(u\nabla v)+g(u)$，$τv_t=Δv-v+u$的高维拟线性抛物-抛物Keller-Segel方程组，其中$Ω$是$\mathbb{R}^n$中的光滑有界区域，$n\ge 2$，$ϕ$和$g$是满足$ks^p\leϕ$的光滑正函数，当$s\ge s_0>1$，$s>0$的$g(s) \le as - μs^2$，带有$g(0)\ge0$和常量$a\ge 0$、$τ,χ,μ>0$。已知无Logistic源的模型同时存在有界解和无界解，通过临界指数$\frac{2}{n}$来确定。另一方面，该模型只是一个Logistic阻尼和聚集效应相互平衡的临界情形，其解的性质由所涉及的系数决定。本文证明了存在$θ_0>0$使得当$\fracχμ<θ_0$满足时，无论初始数据和扩散的大小如何，问题都存在整体有界经典解.这表明了逻辑源对解的行为的实质性影响。

英文摘要：

This paper deals with the higher dimension quasilinear parabolic-parabolic Keller-Segel system involving a source term of logistic type $ u_t=\nabla\cdot(ϕ(u)\nabla u)-χ\nabla\cdot(u\nabla v)+g(u)$, $τv_t=Δv-v+u$ in $Ω\times (0,T)$, subject to nonnegative initial data and homogeneous Neumann boundary condition, where $Ω$ is smooth and bounded domain in $\mathbb{R}^n$, $n\ge 2$, $ϕ$ and $g$ are smooth and positive functions satisfying $ks^p\leϕ$ when $s\ge s_0>1$, $g(s) \le as - μs^2$ for $s>0$ with $g(0)\ge0$ and constants $a\ge 0$, $τ,χ,μ>0$. It was known that the model without the logistic source admits both bounded and unbounded solutions, identified via the critical exponent $\frac{2}{n}$. On the other hand, the model is just a critical case with the balance of logistic damping and aggregation effects, for which the property of solutions should be determined by the coefficients involved. In the present paper it is proved that there is $θ_0>0$ such that the problem admits global bounded classical solutions, regardless of the size of initial data and diffusion whenever $\fracχμ<θ_0$. This shows the substantial effect of the logistic source to the behavior of solutions.

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