# American Institute of Mathematical Sciences

• Previous Article
Parabolic and elliptic problems with general Wentzell boundary condition on Lipschitz domains
• CPAA Home
• This Issue
• Next Article
An infinite dimensional bifurcation problem with application to a class of functional differential equations of neutral type
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:
 [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

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
 [1] Johannes Lankeit. Infinite time blow-up of many solutions to a general quasilinear parabolic-elliptic Keller-Segel system. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 233-255. doi: 10.3934/dcdss.2020013 [2] Tian Xiang. Dynamics in a parabolic-elliptic chemotaxis system with growth source and nonlinear secretion. Communications on Pure & Applied Analysis, 2019, 18 (1) : 255-284. doi: 10.3934/cpaa.2019014 [3] Yūki Naito, Takasi Senba. Oscillating solutions to a parabolic-elliptic system related to a chemotaxis model. Conference Publications, 2011, 2011 (Special) : 1111-1118. doi: 10.3934/proc.2011.2011.1111 [4] Yilong Wang, Xuande Zhang. On a parabolic-elliptic chemotaxis-growth system with nonlinear diffusion. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 321-328. doi: 10.3934/dcdss.2020018 [5] Giuseppe Maria Coclite, Helge Holden, Kenneth H. Karlsen. Wellposedness for a parabolic-elliptic system. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 659-682. doi: 10.3934/dcds.2005.13.659 [6] Rachidi B. Salako, Wenxian Shen. Spreading speeds and traveling waves of a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^N$. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6189-6225. doi: 10.3934/dcds.2017268 [7] Tomasz Cieślak, Kentarou Fujie. Global existence in the 1D quasilinear parabolic-elliptic chemotaxis system with critical nonlinearity. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 165-176. doi: 10.3934/dcdss.2020009 [8] Hua Chen, Wenbin Lv, Shaohua Wu. A free boundary problem for a class of parabolic-elliptic type chemotaxis model. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2577-2592. doi: 10.3934/cpaa.2018122 [9] Yūki Naito, Takasi Senba. Blow-up behavior of solutions to a parabolic-elliptic system on higher dimensional domains. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3691-3713. doi: 10.3934/dcds.2012.32.3691 [10] Kentarou Fujie, Takasi Senba. Global existence and boundedness in a parabolic-elliptic Keller-Segel system with general sensitivity. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 81-102. doi: 10.3934/dcdsb.2016.21.81 [11] Tobias Black. Global generalized solutions to a parabolic-elliptic Keller-Segel system with singular sensitivity. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 119-137. doi: 10.3934/dcdss.2020007 [12] Mengyao Ding, Sining Zheng. $L^γ$-measure criteria for boundedness in a quasilinear parabolic-elliptic Keller-Segel system with supercritical sensitivity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 2971-2988. doi: 10.3934/dcdsb.2018295 [13] Jong-Shenq Guo, Satoshi Sasayama, Chi-Jen Wang. Blowup rate estimate for a system of semilinear parabolic equations. Communications on Pure & Applied Analysis, 2009, 8 (2) : 711-718. doi: 10.3934/cpaa.2009.8.711 [14] Mihaela Negreanu, J. Ignacio Tello. On a Parabolic-ODE system of chemotaxis. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 279-292. doi: 10.3934/dcdss.2020016 [15] Min Zou, An-Ping Liu, Zhimin Zhang. Oscillation theorems for impulsive parabolic differential system of neutral type. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2351-2363. doi: 10.3934/dcdsb.2017103 [16] Sachiko Ishida. $L^\infty$-decay property for quasilinear degenerate parabolic-elliptic Keller-Segel systems. Conference Publications, 2013, 2013 (special) : 335-344. doi: 10.3934/proc.2013.2013.335 [17] Ansgar Jüngel, Oliver Leingang. Blow-up of solutions to semi-discrete parabolic-elliptic Keller-Segel models. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4755-4782. doi: 10.3934/dcdsb.2019029 [18] Liangchen Wang, Yuhuan Li, Chunlai Mu. Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 789-802. doi: 10.3934/dcds.2014.34.789 [19] Wei Wang, Yan Li, Hao Yu. Global boundedness in higher dimensions for a fully parabolic chemotaxis system with singular sensitivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3663-3669. doi: 10.3934/dcdsb.2017147 [20] Bao-Zhu Guo, Liang Zhang. Local exact controllability to positive trajectory for parabolic system of chemotaxis. Mathematical Control & Related Fields, 2016, 6 (1) : 143-165. doi: 10.3934/mcrf.2016.6.143

2018 Impact Factor: 0.925