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September  2013, 12(5): 1861-1880. doi: 10.3934/cpaa.2013.12.1861

Bounded and unbounded oscillating solutions to a parabolic-elliptic system in two dimensional space

1. 

Department of Mathematics, Faculty of Sciences, Ehime University, Matsuyama, 790-8577

2. 

Department of Mathematics, Kyushu Institute of Technology, Sensuicho, Tobata, Kitakyushu 804-8550

Received  March 2011 Revised  October 2012 Published  January 2013

In this paper, we consider solutions to a Cauchy problem for a parabolic-elliptic system in two dimensional space. This system is a simplified version of a chemotaxis model, and is also a model of self-interacting particles.
The behavior of solutions to the problem closely depends on the $L^1$-norm of the solutions. If the quantity is larger than $8\pi$, the solution blows up in finite time. If the quantity is smaller than the critical mass, the solution exists globally in time. In the critical case, infinite blowup solutions were found.
In the present paper, we direct our attention to radial solutions to the problem whose $L^1$-norm is equal to $8\pi$ and find bounded and unbounded oscillating solutions.
Citation: Yūki Naito, Takasi Senba. Bounded and unbounded oscillating solutions to a parabolic-elliptic system in two dimensional space. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1861-1880. doi: 10.3934/cpaa.2013.12.1861
References:
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P. Biler and N. Nadzieja, Existence and nonexistence of solutions for a model of gravitational interaction particles I,, Colloq. Math., 66 (1994), 319. Google Scholar

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P. Biler, G. Karch, P. Laurençot and T. Nadzieja, The $8\pi$ problem for radially symmetric solutions of a chemotaxis model in the plane,, Math. Meth. Appl. Sci., 29 (2006), 1563. doi: 10.1002/mma.743. Google Scholar

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J. Dolbeault and B. Perthame, Optimal critical mass in the two-dimensional Keller-Segel model in $\mathbfR^2$,, C. R. Math. Acad. Sci. Paris, 339 (2004), 611. doi: 10.1016/j.crma.2004.08.011. Google Scholar

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C. Gui, W.-M. Ni and X. Wang, On the stability and instability of positive steady states of a semilinear heat equation in $\mathbfR^n$,, Comm. Pure Appl. Math., 45 (1992), 1153. doi: 10.1002/cpa.3160450906. Google Scholar

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C. Gui, W.-M. Ni and X. Wang, Further study on a nonlinear heat equation,, J. Diff. Eqs., 169 (2001), 588. doi: 10.1006/jdeq.2000.3909. Google Scholar

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E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399. doi: 10.1016/0022-5193(70)90092-5. Google Scholar

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P. Poláčik and E. Yanagida, On bounded and unbounded global solutions of a supercritical semilinear heat equation,, Math. Ann., 337 (2003), 745. Google Scholar

show all references

References:
[1]

P. Biler and N. Nadzieja, Existence and nonexistence of solutions for a model of gravitational interaction particles I,, Colloq. Math., 66 (1994), 319. Google Scholar

[2]

P. Biler, G. Karch, P. Laurençot and T. Nadzieja, The $8\pi$ problem for radially symmetric solutions of a chemotaxis model in the plane,, Math. Meth. Appl. Sci., 29 (2006), 1563. doi: 10.1002/mma.743. Google Scholar

[3]

A. Blanchet, J. A. Carrillo and N. Masmoudi, Infinite time aggregation for the critical two-dimensional Patlak-Keller-Segel model,, Comm. Pure Appl. Math., 61 (2008), 1449. doi: 10.1002/cpa.20225. Google Scholar

[4]

S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis,, Math. Biosci., 56 (1981), 217. doi: 10.1016/0025-5564(81)90055-9. Google Scholar

[5]

J. Dolbeault and B. Perthame, Optimal critical mass in the two-dimensional Keller-Segel model in $\mathbfR^2$,, C. R. Math. Acad. Sci. Paris, 339 (2004), 611. doi: 10.1016/j.crma.2004.08.011. Google Scholar

[6]

E. Feireisl, Ph. Laurençot and H. Petzeltová, On convergence to equilibria for the Keller-Segel chemotaxis model,, J. Differential Equations, 236 (2007), 551. Google Scholar

[7]

C. Gui, W.-M. Ni and X. Wang, On the stability and instability of positive steady states of a semilinear heat equation in $\mathbfR^n$,, Comm. Pure Appl. Math., 45 (1992), 1153. doi: 10.1002/cpa.3160450906. Google Scholar

[8]

C. Gui, W.-M. Ni and X. Wang, Further study on a nonlinear heat equation,, J. Diff. Eqs., 169 (2001), 588. doi: 10.1006/jdeq.2000.3909. Google Scholar

[9]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399. doi: 10.1016/0022-5193(70)90092-5. Google Scholar

[10]

T. Nagai, Global existence and decay estimates of solutions to a parabolic-elliptic system of a drift-diffusion type in $\mathbfR^2$,, Differential Integral Equations, 24 (2011), 29. Google Scholar

[11]

T. Ogawa and T. Nagai, Global existence of solutions to a parabolic-elliptic system of a drift-diffusion type in $\mathbfR^2$,, preprint., (). Google Scholar

[12]

P. Poláčik and E. Yanagida, On bounded and unbounded global solutions of a supercritical semilinear heat equation,, Math. Ann., 337 (2003), 745. Google Scholar

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