论文标题
趋化限制的趋化系统的爆破现象
Blow-up phenomena for a chemotaxis system with flux limitation
论文作者
论文摘要
在本文中，我们考虑了以下抛物线 - 椭圆形交叉扩散系统的非负解决方案 \ begin {equination*} \ left \ {\ begin {array} {l} \ begin {aligned}＆u_t =ΔU-\ nabla（u f（| \ nabla v |^2）\ nabla v） \ frac 1 {|ω|} \int_Ωu dx，\\ [6pt]＆u（x，0）= u_0（x），\ end {aligned} \ end} \ end {array} \ right。 \ end {equation*} in $ω\ times（0，\ infty）$，在$ \ mathbb {r}^n $中，$ω$ a Ball in Ball in Ball in Ball in Neumann Boundary条件下，$ n \ geq 3 $，$ f（$ f（ξ） \ frac {n-2} {2（n-1）} $，它描述了交叉扩散通量的梯度依赖性限制。在$ f $和初始数据的条件下，我们证明了在有限时间内以$ l^\ infty $ -norm爆炸的解决方案，在$ l^p $ -norm中也以某些$ p> 1 $炸毁。此外，得出了爆炸时间的下限。 \ vskip.2truecm \ noindent {\ bf ams主题分类} {primary：35b44;次要：35Q92，92C17。} \ vskip.2Truecm \ noIndent {\ bf关键词：}有限的时间爆炸;趋化性。
In this paper we consider nonnegative solutions of the following parabolic-elliptic cross-diffusion system \begin{equation*} \left\{ \begin{array}{l} \begin{aligned} &u_t = Δu - \nabla(u f(|\nabla v|^2 )\nabla v), \\[6pt] &0= Δv -μ+ u , \quad \int_Ωv =0, \ \ μ:= \frac 1 {|Ω|} \int_Ω u dx, \\[6pt] &u(x,0)= u_0(x), \end{aligned} \end{array} \right. \end{equation*} in $Ω\times (0,\infty)$, with $Ω$ a ball in $\mathbb{R}^N$, $N\geq 3$ under homogeneous Neumann boundary conditions and $f(ξ) = (1+ ξ)^{-α}$, $0<α< \frac{N-2}{2(N-1)}$, which describes gradient-dependent limitation of cross diffusion fluxes. Under conditions on $f$ and initial data, we prove that a solution which blows up in finite time in $L^\infty$-norm, blows up also in $L^p$-norm for some $p>1$. Moreover, a lower bound of blow-up time is derived. \vskip.2truecm \noindent{\bf AMS Subject Classification }{Primary: 35B44; Secondary: 35Q92, 92C17.} \vskip.2truecm \noindent{\bf Key Words:} finite-time blow-up; chemotaxis.
