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2015, 2015(special): 635-643. doi: 10.3934/proc.2015.0635

An iterative approach to $L^\infty$-boundedness in quasilinear Keller-Segel systems

Sachiko Ishida 1,

1. 

Department of Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601

Received  September 2014 Revised  February 2015 Published  November 2015

This paper mainly considers the uniform bound on solutions of non-degenerate Keller-Segel systems on the whole space. In the case that the domain is bounded, Tao-Winkler (2012) proved existence of globally bounded solutions of non-degenerate systems. More precisely, they gave the result on boundedness in quasilinear parabolic equations by using the $L^p$-bounds on the solution for some large $p>1$. In Ishida-Yokota (2012), they dealt with the same system as this paper on the whole space, however, their $L^\infty$-estimate possibly grows up even if the solutions have the uniform bounds in $L^p(\mathbb{R}^N)$ for all $p\in[1,\infty)$. The present work asserts the uniform in time $L^\infty$-bound on solutions. Moreover, this paper covers the degenerate Keller-Segel systems and constructs the uniformly bounded weak solutions.
Keywords: Boundedness, iteration, quasilinear, Keller-Segel system, degenerate..
Mathematics Subject Classification: Primary: 35K51; Secondary: 35D30, 35Q32, 92C1.
Citation: Sachiko Ishida. An iterative approach to $L^\infty$-boundedness in quasilinear Keller-Segel systems. Conference Publications, 2015, 2015 (special) : 635-643. doi: 10.3934/proc.2015.0635
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show all references

References:
[1]

J. Differential Equations, 252 (2012), 5832-5851.  Google Scholar

[2]

Acta Appl. Math., 129 (2014), 135-146.  Google Scholar

[3]

Ann. Scuola Norm. Sup. Pisa Cl. Sci., (4) 24 (1997), 633-683.  Google Scholar

[4]

J. Math. Biol., 58 (2009), 183-217.  Google Scholar

[5]

Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2537-2568.  Google Scholar

[6]

J. Differential Equations, 252 (2012), 2469-2491.  Google Scholar

[7]

Discrete Contin. Dyn. Syst. Supplements, 2013 (2013), 345-354. Google Scholar

[8]

Discrete Contin. Dyn. Syst. Ser. B, 18 (2013) 2569-2596.  Google Scholar

[9]

J. Theor. Biol., 26 (1970), 399-415. Google Scholar

[10]

Nonlinear Anal., 47 (2001), 777-787.  Google Scholar

[11]

Funkcial. Ekvac., 40 (1997), 411-433.  Google Scholar

[12]

Funkcial. Ekvac., 46 (2003), 383-407.  Google Scholar

[13]

Funkcial. Ekvac., 44 (2001), 441-469.  Google Scholar

[14]

Mathematical Surveys and Monographs, 49 American Mathematical Society, Providence, RI, (1997). Google Scholar

[15]

J. Differential Equations, 227 (2006), 333-364.  Google Scholar

[16]

J. Differential Equations, 250 (2011), 3047-3087.  Google Scholar

[17]

J. Differential Equations, 252 (2012), 692-715.  Google Scholar

[18]

Math. Nachr., 283 (2010), 1664-1673.  Google Scholar

[19]

J. Differential Equations, 248 (2010), 2889-2905.  Google Scholar

[20]

J. Math. Pures Appl., 100 (2013), 748-767.  Google Scholar

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