Finite-time blowup in Cauchy problem of parabolic-parabolic chemotaxis system
Finite-time blowup in Cauchy problem of parabolic-parabolic chemotaxis system
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DOI：
10.1016/j.matpur.2019.10.004
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发表时间：
2020-04
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影响因子：
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通讯作者：
N. Mizoguchi
中科院分区：
文献类型：
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作者：
N. Mizoguchi
本文讨论一类描述趋化聚集的抛物-抛物系统的爆破问题。在圆盘中，如果其初始能量小于某一值，则径向解在有限时间内爆炸。在整个平面上，对于任何正向自相似解，随着时间的推移，能量向−∞发散到+∞。这意味着人们不能期望像在圆盘中那样利用能量得到有限时间爆破的充分条件。对于溶液(u，v)，u和v分别表示细胞密度和化学物质密度。设τ是v的时间导数系数，我们首先证明了对于τ>；0，存在M(τ)>；0，其中M(τ)→∞)为τ→∞，使得所有初始质量u大于M(τ)的径向解(u，v)在有限时间内爆破。另一方面，在[22]中证明了τ=1的系统中的任何爆破都是II型的(在径向情况下不一定)。去除对τ的限制，我们得到了所有τ>；0的结论。
This paper is concerned with blowup in a parabolic-parabolic system describing chemotactic aggregation. In a disk, radial solutions blow up in finite time if their initial energy is less than some value. In the whole plane, the energy diverges to−∞ as time goes to+∞ for any forward selfsimilar solution. This implies that one cannot expect to get a sufficient condition for finite-time blowup using energy as in a disk. For a solution (u, v), u and v denote density of cells and of chemical substance, respectively. Let τ be the coefficient of time derivative of v. We first prove that for τ> 0 there exists M (τ)> 0 with M (τ)→∞ as τ→∞ such that all radial solutions (u, v) with initial mass of u larger than M (τ) blow up in finite time. On the other hand, it was shown in [22] that any blowup in the system with τ= 1 is type II (not necessarily in radial case). Removing the restriction on τ, we get the conclusion for all τ> 0.