Chemotactic collapse in a parabolic system of mathematical biology

By T. Nagai, T. Senba, Takashi Suzuki. 2000

184 citations recorded by the source corpus.

In 1970, Keller and Segel proposed a parabolic system describing the chemotactic feature of cellular slime molds and recently, several mathematical works have been devoted to it. In the present paper, we study its blowup mechamism and prove the following. First, chemotactic collapse occurs at each isolated blowup point. Next, any blowup point is isolated, provided that the Lyapunov function is bounded from below. Finally, only the origin can be a blowup point of radially symmetric solutions.

Subfields: Biomedical Engineering, Modeling and Simulation, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/f29223ebbd72d54c5a659bb96722fbc2f97b931e

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In Biomedical Engineering

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Citations in the corpus

On the parabolic-elliptic Patlak-Keller-Segel system in dimension 2 and higher (2011, background): “Such a solution develop Dirac masses of mass exactly egal to 8π, see [49, 33].”
Parabolic System of Chemotaxis: Blowup in a Finite and the Infinite Time (2001, background): “6) is proven when the Lyapunov function W is bounded, or u and v are radially symmetric, or A = 47r ([17], [9]).”
The drift–diffusion system in two-dimensional critical Hardy space (2008, unknown): “[1,28,29,41–44,46,47,51–53,56]).”
Blowup of nonradial solutions to parabolic–elliptic systems modeling chemotaxis in two-dimensional domains (2001, background): “that in [ 23 ,27] finite-time blowup in two dimensions necessarily leads to”
Global existence for the Cauchy problem of the parabolic–parabolic Keller–Segel system on the plane (2012, unknown): “3 in [20], which treated the Cauchy-Neumann problem in a bounded domain.”
Global existence, boundedness and stabilization in a high-dimensional chemotaxis system with consumption (2016, background): “Even more delicate results pertaining to the precise structure of the singularity formation ([5], [16]), which occurs in the form of Dirac-masses, or the type of blow-up ([14]) have been achieved.”