# American Institute of Mathematical Sciences

March  2017, 37(3): 1603-1630. doi: 10.3934/dcds.2017066

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

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

Received  July 2014 Revised  October 2016 Published  December 2016

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.

Citation: Yoshifumi Mimura. Critical mass of degenerate Keller-Segel system with no-flux and Neumann boundary conditions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1603-1630. doi: 10.3934/dcds.2017066
##### References:
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##### References:
 [1] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, 2000. Google Scholar [2] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures Lectures in Mathematics, Birkhäuser, 2005. Google Scholar [3] L. Ambrosio and G. Savaré, Gradient flows of probability measures, Handbook of differential equations: Evolutionary equations, Handb. Differ. Equ. , Elsevier/North-Holland, Amsterdam, Ⅲ (2007), 1–136. Google Scholar [4] J. Bedrossian, N. Rodríguez and A. Bertozzi, Local and global well-posedness for aggregation equations and Patlak-Keller-Segel models with degenerate diffusion, Nonlinearity, 24 (2011), 1683-1714. Google Scholar [5] P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743. Google Scholar [6] A. Blanchet, J. A. Carrillo and Ph. Laurençot, Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions, Calc. Var. Partial Differential Equations, 35 (2009), 133-168. Google Scholar [7] A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions, Electron. J. Differential Equations 44 (2006), 32 pp. (electronic). Google Scholar [8] A. Blanchet and Ph. Laurençcot, The Parabolic-Parabolic Keller-Segel System with Critical Diffusion as a Gradient Flow in $\mathbb{R}^d, d ≥q 3$, Comm. Partial Differential Equations, 38 (2013), 658-686. Google Scholar [9] V. Calvez and L. Corrias, The parabolic-parabolic Keller-Segel model in $\mathbb{R}^2$, Commun. Math. Sci., 6 (2008), 417-447. Google Scholar [10] H. Gajewski and K. Zacharias, Global behavior of a reaction-diffusion system modeling chemotaxis, Math. Nachr., 195 (1998), 77-114. Google Scholar [11] M. A. Herrero and J. J. L. Velázquez, Chemotaxis collapse for Keller-Segel model, J. Math. Biol., 35 (1996), 177-194. Google Scholar [12] M. A. Herrero and J. J. L. Velázquez, Singularity patterns in a chemotaxis model, Math. Ann., 306 (1996), 583-623. Google Scholar [13] S. Ishida and T. Yokota, Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2569-2596. Google Scholar [14] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. Google Scholar [15] D. Matthes, R. J. McCann and G. Savarß, A family of nonlinear fourth order equations of gradient flow type, Comm. Part. Diff. Eqs., 34 (2009), 1352-1397. Google Scholar [16] N. Mizoguchi, Global existence for the Cauchy problem of the parabolic-parabolic Keller-Segel system on the plane, Calc. Var. Partial Differential Equations, 48 (2013), 491-505. Google Scholar [17] T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433. Google Scholar [18] F. Otto, Doubly degenerate diffusion equations as steepest descent, Manuscript, 1996.Google Scholar [19] T. Suzuki and R. Takahashi, Degenerate parabolic equation with critical exponent derived from the kinetic theory, Ⅱ. Blowup threshold, Differential Integral Equations, 22 (2009), 1153-1172. Google Scholar [20] M. Taylor, Partial Differential Equations I Springer New York, 1996. Google Scholar [21] C. Villani, Topics in Optimal Transportation Graduate Studies in Mathematics 58, American Mathematical Society, Providence, RI, 2003. Google Scholar
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