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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

Tomasz Cieślak 1,

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 $\Omega$.
Keywords: Chemotaxis..
Mathematics Subject Classification: 35B44, 35K20, 35K55, 92C1.
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), 201-225. 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), 715-743. 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), 1189-1209. 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), 319-334.  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), 5215-5228. 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), 633-683.  Google Scholar

[8]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biology, 26 (1970), 399-415. 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), 1077-1092.  Google Scholar

[10]

T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.  Google Scholar

[11]

T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55. 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), 411-433.  Google Scholar

[13]

V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology, J. Theoretical Biology, 42 (1973), 63-105. 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), 473-483.  Google Scholar

[15]

V. I. Yudovich, Some estimates connected with integral operators and with solutions of elliptic equations, Dokl. Akad. Nauk SSSR, 138 (1961), 805-808.  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), 201-225. 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), 715-743. 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), 1189-1209. 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), 319-334.  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), 5215-5228. 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), 633-683.  Google Scholar

[8]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biology, 26 (1970), 399-415. 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), 1077-1092.  Google Scholar

[10]

T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.  Google Scholar

[11]

T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55. 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), 411-433.  Google Scholar

[13]

V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology, J. Theoretical Biology, 42 (1973), 63-105. 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), 473-483.  Google Scholar

[15]

V. I. Yudovich, Some estimates connected with integral operators and with solutions of elliptic equations, Dokl. Akad. Nauk SSSR, 138 (1961), 805-808.  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

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