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

We consider a Cauchy problem for a parabolic-elliptic system of drift–diffusion type. The problem is formally of the form This system describes a mass-conserving aggregation phenomenon including gravitational collapse and bacterial chemotaxis. Our concern is the asymptotic behavior of blowup solutio...

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Bibliographic Details
Published in:Archive for rational mechanics and analysis 2011-08, Vol.201 (2), p.549-573
Main Authors: Giga, Yoshikazu, Mizoguchi, Noriko, Senba, Takasi
Format: Article
Language:English
Subjects:
Bacteriology
Biological and medical sciences
Classical Mechanics
Complex Systems
Exact sciences and technology
Fluid- and Aerodynamics
Fundamental and applied biological sciences. Psychology
Mathematical analysis
Mathematical and Computational Physics
Mathematics
Microbiology
Motility, taxis
Ordinary differential equations
Partial differential equations
Physics
Physics and Astronomy
Sciences and techniques of general use
Theoretical
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Summary:We consider a Cauchy problem for a parabolic-elliptic system of drift–diffusion type. The problem is formally of the form 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.
ISSN:0003-9527
1432-0673
DOI:10.1007/s00205-010-0394-7