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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

Yoshifumi Mimura

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.

Keywords: Keller-Segel, chemotaxis, degenerate diffusion, Wasserstein distance, gradient flows.
Mathematics Subject Classification: 35K65, 35B33, 47J30, 35K40.
Citation: Yoshifumi Mimura. Critical mass of degenerate Keller-Segel system with no-flux and Neumann boundary conditions. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1603-1630. doi: 10.3934/dcds.2017066
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

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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

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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

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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

show all references

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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