论文标题
凯勒（Keller）解决方案的行为 - 具有通量限制和源术语的识别系统
Behavior in time of solutions of a Keller--Segel system with flux limitation and source term
论文作者
论文摘要
在本文中，我们考虑了以下抛物线式交叉扩散系统的径向对称解\ begin {equation*} \ begin {case} u_t =ΔU-\ nabla \ cdot（u f（| \ nabla v |^2）\ nabla v |^2） \int_Ωv\，dx = 0，＆\\ [2mm] u（x，0）= u_0（x），＆\ end {cases} \ end {eque} \ end {equication*} in $ω\ times（0，\ infty）$，$ \ ne n $ ne $ ne n $ ne $ ne $ ne \ ne \ ne \ n $ ne \ ne \ ne， $g(u)= λu - μu^k$ , $λ>0, \ μ>0$, and $ k >1$, $f(|\nabla v|^2 )= k_f(1+ |\nabla v|^2)^{-α}$, $α>0$, which describes gradient-dependent limitation of cross diffusion fluxes.功能$ m（t）$是$ u（x，t）$的时间依赖的空间平均值，即$ m（t）：= \ frac 1 {|ω|} \int_ΩU（x，x，t）\，dx $。在$α$和$ k $的较小条件下，我们证明解决方案$ u（x，t）$以$ l^{\ infty} $吹出，在有限的时间下$ t_ {max {max} $，对于一些$ p> 1 $，它也以$ l^p $ -norm炸开。另外，得出了爆破时间的下限。最后，在$α$或$ k $的巨大条件下，我们证明该解决方案是全球性的并且是及时的。
In this paper we consider radially symmetric solutions of the following parabolic--elliptic cross-diffusion system \begin{equation*} \begin{cases} u_t = Δu - \nabla \cdot (u f(|\nabla v|^2 )\nabla v) + g(u), & \\[2mm] 0= Δv -m(t)+ u , \quad \int_Ωv \,dx=0, & \\[2mm] u(x,0)= u_0(x), & \end{cases} \end{equation*} in $Ω\times (0,\infty)$, with $Ω$ a ball in $\mathbb{R}^N$, $N\geq 3$, under homogeneous Neumann boundary conditions, where $g(u)= λu - μu^k$ , $λ>0, \ μ>0$, and $ k >1$, $f(|\nabla v|^2 )= k_f(1+ |\nabla v|^2)^{-α}$, $α>0$, which describes gradient-dependent limitation of cross diffusion fluxes. The function $m(t)$ is the time dependent spatial mean of $u(x,t)$ i.e. $m(t) := \frac 1 {|Ω|} \int_Ω u(x,t) \,dx$. Under smallness conditions on $α$ and $k$, we prove that the solution $u(x,t)$ blows up in $L^{\infty}$-norm at finite time $T_{max}$ and for some $p>1$ it blows up also in $L^p$-norm. In addition a lower bound of blow-up time is derived. Finally, under largeness conditions on $α$ or $k$, we prove that the solution is global and bounded in time.
