Asymptotic Behavior of Type I Blowup Solutions to a Parabolic-Elliptic System of Drift-Diffusion Type
Asymptotic Behavior of Type I Blowup Solutions to a Parabolic-Elliptic System of Drift-Diffusion Type
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DOI：
10.1007/s00205-010-0394-7
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发表时间：
2011-08-01
影响因子：
2.5
通讯作者：
Senba, Takasi
中科院分区：
文献类型：
--
作者：
Giga, Yoshikazu;Mizoguchi, Noriko;Senba, Takasi
本文考虑一类漂移扩散型抛物-椭圆方程组的柯西问题。该问题的形式为U（t）= del。(del U - U del（-Delta）（-1）U）。该系统描述了一种质量守恒的聚集现象，包括引力坍缩和细菌的趋化性。我们关注的是当爆破是第一类时爆破解的渐近行为，即爆破速率与相应的常微分方程y（t）= y（2）相同（直到一个倍数常数）.证明了当爆破集是单点集且空间维数大于等于3时，只要解是径向非负的，所有的I型爆破都是渐近（向后）自相似的.
We consider a Cauchy problem for a parabolic-elliptic system of drift-diffusion type. The problem is formally of the form U(t) = del . (del U - U del(-Delta)(-1)U). This system describes a mass-conserving aggregation phenomenon including gravitational collapse and bacterial chemotaxis. Our concern is the asymptotic behavior of blowup solutions when the blowup is type I, in the sense that its blowup rate is the same as the corresponding ordinary differential equation y (t) = y (2) (up to a multiple constant). It is shown that all type I blowup is asymptotically (backward) self-similar, provided that the solution is radial, nonnegative when the blowup set is a singleton and the space dimension is greater than or equal to three.