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
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show all references

References:
[1]

J. Differential Equations, 33 (1979), 201-225. doi: 10.1016/0022-0396(79)90088-3.  Google Scholar

[2]

Adv. Math. Sci. Appl., 8 (1998), 715-743. Google Scholar

[3]

Nonlinear Anal. TMA, 23 (1994), 1189-1209. doi: 10.1016/0362-546X(94)90101-5.  Google Scholar

[4]

Colloq. Math., 66 (1994), 319-334.  Google Scholar

[5]

Nonlinear Anal. TMA, 75 (2012), 5215-5228. doi: 10.1016/j.na.2012.04.038.  Google Scholar

[6]

Acta Appl. Math., (2013). doi: 10.1007/s10440-013-9832-5.  Google Scholar

[7]

Ann. Scuola Norm. Super. Pisa Cl. Sci., 24 (1997), 633-683.  Google Scholar

[8]

J. Theor. Biology, 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[9]

Indiana Univ. Math. J., 20 (1971), 1077-1092.  Google Scholar

[10]

Adv. Math. Sci. Appl., 5 (1995), 581-601.  Google Scholar

[11]

J. Inequal. Appl., 6 (2001), 37-55. doi: 10.1155/S1025583401000042.  Google Scholar

[12]

Funkc. Ekvacioj, 40 (1997), 411-433.  Google Scholar

[13]

J. Theoretical Biology, 42 (1973), 63-105. doi: 10.1016/0022-5193(73)90149-5.  Google Scholar

[14]

Indiana Univ. Math. J., 17 (1967), 473-483.  Google Scholar

[15]

Dokl. Akad. Nauk SSSR, 138 (1961), 805-808.  Google Scholar

[16]

J. Math. Pures Appl., (2013). doi: 10.1016/j.matpur.2013.01.020.  Google Scholar

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