Asymptotic Behavior of Type I Blowup Solutions to a Parabolic-Elliptic System of Drift-Diffusion Type

作者：Giga Yoshikazu^*; Mizoguchi Noriko; Senba Takasi

来源：Archive for Rational Mechanics and Analysis, 2011, 201(2): 549-573.

DOI：10.1007/s00205-010-0394-7

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.

• 出版日期2011-8