论文标题
在两种趋化性模型中，趋化效应是否会导致爆炸吗？
Can chemotactic effects lead to blow-up or not in two-species chemotaxis-competition models?
论文作者
论文摘要
本文涉及两种类型的趋化性模型\ begin {align*} \ begin {case} u_t =d_1Δu -χ_1\ nabla \ cdot（u \ nabla w） +μ_1u（1- u^{κ_1-1} -a_1 v^{λ_1-1}）， ＆\ quad x \ inω，\ t> 0，\\％ v_t =D_2ΔV - χ_2\ nabla \ cdot（v \ nabla w） +μ_2V（1- a_2 u^{λ_2-1} -v^{κ_2-1}）， ＆\ quad x \ inω，\ t> 0，\\％ 0 =d_3Δw +αu +βv -h（u，v，w）， ＆\ quad x \ inω，\ t> 0， \ end {cases} \ end {align*}其中$ω\ subset \ mathbb {r}^n $ $（n \ ge2）$是一个具有光滑边界的有界域，$ h =γw$或$ h = \ frac {1} {1} {|ω|} {|ω| |} \ int_ \int_Ω（αU+βV）在$κ_1=λ_1=κ_2=λ_2= 2 $和$ h =γw$的情况下，众所周知，趋化效应的较小条件会导致溶液的界限（MATH。\ MADEDS APPL。\ SCI。\ SCI。; 2018; 41; 41; 41; 234--249）。但是，似乎还没有研究趋化作用很大的情况。因此，在趋化效应很大的情况下，溶液是否也有界，仍然要考虑一个问题。本文的目的是对这个问题给出负面答案。
This paper deals with the two-species chemotaxis-competition models \begin{align*} \begin{cases} u_t = d_1 Δu - χ_1 \nabla \cdot (u \nabla w) + μ_1 u (1- u^{κ_1-1} - a_1 v^{λ_1-1}), &\quad x \in Ω,\ t>0,\\ % v_t = d_2 Δv - χ_2 \nabla \cdot (v \nabla w) + μ_2 v (1- a_2 u^{λ_2-1} - v^{κ_2-1}), &\quad x \in Ω,\ t>0,\\ % 0 = d_3 Δw + αu + βv - h(u,v,w), &\quad x \in Ω,\ t>0, \end{cases} \end{align*} where $Ω\subset \mathbb{R}^n$ $(n\ge2)$ is a bounded domain with smooth boundary, and $h=γw$ or $h=\frac{1}{|Ω|}\int_Ω(αu+ βv)\,dx$. In the case that $κ_1=λ_1=κ_2=λ_2=2$ and $h=γw$, it is known that smallness conditions for the chemotacic effects lead to boundedness of solutions (Math.\ Methods Appl.\ Sci.; 2018; 41; 234--249). However, the case that the chemotactic effects are large seems not to have been studied yet; therefore it remains to consider the question whether the solution is bounded also in the case that the chemotactic effects are large. The purpose of this paper is to give a negative answer to this question.
