Critical mass of degenerate Keller-Segel system with no-flux and Neumann boundary conditions

Yoshifumi Mimura

doi:10.3934/dcds.2017066

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2017, Volume 37, Issue 3: 1603-1630. Doi: 10.3934/dcds.2017066

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Critical mass of degenerate Keller-Segel system with no-flux and Neumann boundary conditions

Yoshifumi Mimura

Graduate School of Mathematical Sciences, The University of Tokyo, Komaba, Tokyo 153-8914, Japan

Received: June 30, 2014

Revised: September 30, 2016

Published: May 2017

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Abstract

We prove the existence of solutions of degenerate parabolic-parabolic Keller-Segel system with no-flux and Neumann boundary conditions for each variable respectively, under the assumption that the total mass of the first variable is below a certain constant. The proof relies on the interpretation of the system as a gradient flow in the product space of the Wasserstein space and the standard $L^2$-space. More precisely, we apply the ''minimizing movement'' scheme and show a certain critical mass appears in the application of this scheme to our problem.

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Mathematics Subject Classification: 35K65, 35B33, 47J30, 35K40.

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