December  2013, 18(10): 2505-2512. doi: 10.3934/dcdsb.2013.18.2505

Trudinger-Moser type inequality for radially symmetric functions in a ring and applications to Keller-Segel in a ring

1. 

Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-956 Warszawa, Poland

Received  April 2013 Revised  July 2013 Published  October 2013

We prove that for radially symmetric functions in a ring $\Omega = ${$ x \in \mathbb{R}^n, n \geq 2 : r \leq |x| \leq R $} a special type of Trudinger-Moser-like inequality holds. Next we show how to infer from it a lack of blowup of radially symmetric solutions to a Keller-Segel system in $\Omega$.
Citation: Tomasz Cieślak. Trudinger-Moser type inequality for radially symmetric functions in a ring and applications to Keller-Segel in a ring. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2505-2512. doi: 10.3934/dcdsb.2013.18.2505
References:
[1]

N. Alikakos, An Application of the Invariance Principle to Reaction-Diffusion Equations,, J. Differential Equations, 33 (1979), 201.  doi: 10.1016/0022-0396(79)90088-3.  Google Scholar

[2]

P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis,, Adv. Math. Sci. Appl., 8 (1998), 715.   Google Scholar

[3]

P. Biler, W. Hebisch and T. Nadzieja, The Debye system: Existence and large time behavior of solutions,, Nonlinear Anal. TMA, 23 (1994), 1189.  doi: 10.1016/0362-546X(94)90101-5.  Google Scholar

[4]

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

[5]

J. Burczak, T. Cieślak and C. Morales-Rodrigo, Global existence vs. Blowup in a fully parabolic quasilinear 1D Keller-Segel system,, Nonlinear Anal. TMA, 75 (2012), 5215.  doi: 10.1016/j.na.2012.04.038.  Google Scholar

[6]

T. Cieślak and C. Stinner, Finite-time blowup in a supercritical quasilinear parabolic-parabolic Keller-Segel system in dimension 2,, Acta Appl. Math., (2013).  doi: 10.1007/s10440-013-9832-5.  Google Scholar

[7]

M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model,, Ann. Scuola Norm. Super. Pisa Cl. Sci., 24 (1997), 633.   Google Scholar

[8]

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

[9]

J. Moser, A sharp form of an inequality of N. Trudinger,, Indiana Univ. Math. J., 20 (1971), 1077.   Google Scholar

[10]

T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system,, Adv. Math. Sci. Appl., 5 (1995), 581.   Google Scholar

[11]

T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains,, J. Inequal. Appl., 6 (2001), 37.  doi: 10.1155/S1025583401000042.  Google Scholar

[12]

T. Nagai, T. Senba, K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis,, Funkc. Ekvacioj, 40 (1997), 411.   Google Scholar

[13]

V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology,, J. Theoretical Biology, 42 (1973), 63.  doi: 10.1016/0022-5193(73)90149-5.  Google Scholar

[14]

N. Trudinger, On imbeddings into Orlicz spaces and some applications,, Indiana Univ. Math. J., 17 (1967), 473.   Google Scholar

[15]

V. I. Yudovich, Some estimates connected with integral operators and with solutions of elliptic equations,, Dokl. Akad. Nauk SSSR, 138 (1961), 805.   Google Scholar

[16]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system,, J. Math. Pures Appl., (2013).  doi: 10.1016/j.matpur.2013.01.020.  Google Scholar

show all references

References:
[1]

N. Alikakos, An Application of the Invariance Principle to Reaction-Diffusion Equations,, J. Differential Equations, 33 (1979), 201.  doi: 10.1016/0022-0396(79)90088-3.  Google Scholar

[2]

P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis,, Adv. Math. Sci. Appl., 8 (1998), 715.   Google Scholar

[3]

P. Biler, W. Hebisch and T. Nadzieja, The Debye system: Existence and large time behavior of solutions,, Nonlinear Anal. TMA, 23 (1994), 1189.  doi: 10.1016/0362-546X(94)90101-5.  Google Scholar

[4]

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

[5]

J. Burczak, T. Cieślak and C. Morales-Rodrigo, Global existence vs. Blowup in a fully parabolic quasilinear 1D Keller-Segel system,, Nonlinear Anal. TMA, 75 (2012), 5215.  doi: 10.1016/j.na.2012.04.038.  Google Scholar

[6]

T. Cieślak and C. Stinner, Finite-time blowup in a supercritical quasilinear parabolic-parabolic Keller-Segel system in dimension 2,, Acta Appl. Math., (2013).  doi: 10.1007/s10440-013-9832-5.  Google Scholar

[7]

M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model,, Ann. Scuola Norm. Super. Pisa Cl. Sci., 24 (1997), 633.   Google Scholar

[8]

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

[9]

J. Moser, A sharp form of an inequality of N. Trudinger,, Indiana Univ. Math. J., 20 (1971), 1077.   Google Scholar

[10]

T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system,, Adv. Math. Sci. Appl., 5 (1995), 581.   Google Scholar

[11]

T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains,, J. Inequal. Appl., 6 (2001), 37.  doi: 10.1155/S1025583401000042.  Google Scholar

[12]

T. Nagai, T. Senba, K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis,, Funkc. Ekvacioj, 40 (1997), 411.   Google Scholar

[13]

V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology,, J. Theoretical Biology, 42 (1973), 63.  doi: 10.1016/0022-5193(73)90149-5.  Google Scholar

[14]

N. Trudinger, On imbeddings into Orlicz spaces and some applications,, Indiana Univ. Math. J., 17 (1967), 473.   Google Scholar

[15]

V. I. Yudovich, Some estimates connected with integral operators and with solutions of elliptic equations,, Dokl. Akad. Nauk SSSR, 138 (1961), 805.   Google Scholar

[16]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system,, J. Math. Pures Appl., (2013).  doi: 10.1016/j.matpur.2013.01.020.  Google Scholar

[1]

Shangbing Ai, Wenzhang Huang, Zhi-An Wang. Reaction, diffusion and chemotaxis in wave propagation. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 1-21. doi: 10.3934/dcdsb.2015.20.1

[2]

Nicolas Vauchelet. Numerical simulation of a kinetic model for chemotaxis. Kinetic & Related Models, 2010, 3 (3) : 501-528. doi: 10.3934/krm.2010.3.501

[3]

Kentarou Fujie, Akio Ito, Michael Winkler, Tomomi Yokota. Stabilization in a chemotaxis model for tumor invasion. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 151-169. doi: 10.3934/dcds.2016.36.151

[4]

Alina Chertock, Alexander Kurganov, Xuefeng Wang, Yaping Wu. On a chemotaxis model with saturated chemotactic flux. Kinetic & Related Models, 2012, 5 (1) : 51-95. doi: 10.3934/krm.2012.5.51

[5]

Hua Chen, Shaohua Wu. The moving boundary problem in a chemotaxis model. Communications on Pure & Applied Analysis, 2012, 11 (2) : 735-746. doi: 10.3934/cpaa.2012.11.735

[6]

Nicola Bellomo, Youshan Tao. Stabilization in a chemotaxis model for virus infection. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 105-117. doi: 10.3934/dcdss.2020006

[7]

Monica Marras, Stella Vernier-Piro, Giuseppe Viglialoro. Decay in chemotaxis systems with a logistic term. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 257-268. doi: 10.3934/dcdss.2020014

[8]

Mihaela Negreanu, J. Ignacio Tello. On a Parabolic-ODE system of chemotaxis. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 279-292. doi: 10.3934/dcdss.2020016

[9]

José Antonio Carrillo, Stefano Lisini, Edoardo Mainini. Uniqueness for Keller-Segel-type chemotaxis models. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1319-1338. doi: 10.3934/dcds.2014.34.1319

[10]

Tomomi Yokota, Noriaki Yoshino. Existence of solutions to chemotaxis dynamics with logistic source. Conference Publications, 2015, 2015 (special) : 1125-1133. doi: 10.3934/proc.2015.1125

[11]

Andriy Sokolov, Robert Strehl, Stefan Turek. Numerical simulation of chemotaxis models on stationary surfaces. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2689-2704. doi: 10.3934/dcdsb.2013.18.2689

[12]

Shubo Zhao, Ping Liu, Mingchao Jiang. Stability and bifurcation analysis in a chemotaxis bistable growth system. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1165-1174. doi: 10.3934/dcdss.2017063

[13]

Anne Nouri, Christian Schmeiser. Aggregated steady states of a kinetic model for chemotaxis. Kinetic & Related Models, 2017, 10 (1) : 313-327. doi: 10.3934/krm.2017013

[14]

Xin Lai, Xinfu Chen, Mingxin Wang, Cong Qin, Yajing Zhang. Existence, uniqueness, and stability of bubble solutions of a chemotaxis model. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 805-832. doi: 10.3934/dcds.2016.36.805

[15]

Piotr Biler, Grzegorz Karch, Jacek Zienkiewicz. Morrey spaces norms and criteria for blowup in chemotaxis models. Networks & Heterogeneous Media, 2016, 11 (2) : 239-250. doi: 10.3934/nhm.2016.11.239

[16]

Raul Borsche, Axel Klar, T. N. Ha Pham. Nonlinear flux-limited models for chemotaxis on networks. Networks & Heterogeneous Media, 2017, 12 (3) : 381-401. doi: 10.3934/nhm.2017017

[17]

Johannes Lankeit. Chemotaxis can prevent thresholds on population density. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1499-1527. doi: 10.3934/dcdsb.2015.20.1499

[18]

T. Hillen, K. Painter, Christian Schmeiser. Global existence for chemotaxis with finite sampling radius. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 125-144. doi: 10.3934/dcdsb.2007.7.125

[19]

Zhi-An Wang. Mathematics of traveling waves in chemotaxis --Review paper--. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 601-641. doi: 10.3934/dcdsb.2013.18.601

[20]

Alexander Kurganov, Mária Lukáčová-Medvidová. Numerical study of two-species chemotaxis models. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 131-152. doi: 10.3934/dcdsb.2014.19.131

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]