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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

 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.
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
##### References:
 [1] T. Cieślak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions,, J. Differential Equations, 252 (2012), 5832.   Google Scholar [2] T. Cieślak and C. Stinner, Finite-time blowup in a supercritical quasilinear parabolic-parabolic Keller-Segel system in dimension 2,, Acta Appl. Math., 129 (2014), 135.   Google Scholar [3] M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 633.   Google Scholar [4] T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183.   Google Scholar [5] S. Ishida, Y. Maeda and T. Yokota, Gradient estimate for solutions to quasilinear non-degenerate Keller-Segel systems on $\mathbbR^N$,, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2537.   Google Scholar [6] S. Ishida and T. Yokota, Global existence of weak solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type with small data,, J. Differential Equations, 252 (2012), 2469.   Google Scholar [7] S. Ishida and T. Yokota, Remarks on the global existence of weak solutions to quasilinear degenerate Keller-Segel systems,, Discrete Contin. Dyn. Syst. Supplements, 2013 (2013), 345.   Google Scholar [8] S. Ishida and T. Yokota, Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type,, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2569.   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.   Google Scholar [10] T. Nagai, Global existence and blowup of solutions to a chemotaxis system,, Nonlinear Anal., 47 (2001), 777.   Google Scholar [11] T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis,, Funkcial. Ekvac., 40 (1997), 411.   Google Scholar [12] T. Nagai, R. Syukuinn and M. Umesako, Decay properties and asymptotic profiles of bounded solutions to a parabolic system of chemotaxis in $\mathbbR^N$,, Funkcial. Ekvac., 46 (2003), 383.   Google Scholar [13] K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations,, Funkcial. Ekvac., 44 (2001), 441.   Google Scholar [14] R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations,, Mathematical Surveys and Monographs, 49 (1997).   Google Scholar [15] Y. Sugiyama and H. Kunii, Global existence and decay properties for a degenerate Keller-Segel model with a power factor in drift term,, J. Differential Equations, 227 (2006), 333.   Google Scholar [16] Y. Sugiyama and Y. Yahagi, Extinction, decay and blow-up for Keller-Segel systems of fast diffusion type,, J. Differential Equations, 250 (2011), 3047.   Google Scholar [17] Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity,, J. Differential Equations, 252 (2012), 692.   Google Scholar [18] M. Winkler, Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity,, Math. Nachr., 283 (2010), 1664.   Google Scholar [19] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model,, J. Differential Equations, 248 (2010), 2889.   Google Scholar [20] M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system,, J. Math. Pures Appl., 100 (2013), 748.   Google Scholar

show all references

##### References:
 [1] T. Cieślak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions,, J. Differential Equations, 252 (2012), 5832.   Google Scholar [2] T. Cieślak and C. Stinner, Finite-time blowup in a supercritical quasilinear parabolic-parabolic Keller-Segel system in dimension 2,, Acta Appl. Math., 129 (2014), 135.   Google Scholar [3] M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 633.   Google Scholar [4] T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183.   Google Scholar [5] S. Ishida, Y. Maeda and T. Yokota, Gradient estimate for solutions to quasilinear non-degenerate Keller-Segel systems on $\mathbbR^N$,, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2537.   Google Scholar [6] S. Ishida and T. Yokota, Global existence of weak solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type with small data,, J. Differential Equations, 252 (2012), 2469.   Google Scholar [7] S. Ishida and T. Yokota, Remarks on the global existence of weak solutions to quasilinear degenerate Keller-Segel systems,, Discrete Contin. Dyn. Syst. Supplements, 2013 (2013), 345.   Google Scholar [8] S. Ishida and T. Yokota, Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type,, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2569.   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.   Google Scholar [10] T. Nagai, Global existence and blowup of solutions to a chemotaxis system,, Nonlinear Anal., 47 (2001), 777.   Google Scholar [11] T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis,, Funkcial. Ekvac., 40 (1997), 411.   Google Scholar [12] T. Nagai, R. Syukuinn and M. Umesako, Decay properties and asymptotic profiles of bounded solutions to a parabolic system of chemotaxis in $\mathbbR^N$,, Funkcial. Ekvac., 46 (2003), 383.   Google Scholar [13] K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations,, Funkcial. Ekvac., 44 (2001), 441.   Google Scholar [14] R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations,, Mathematical Surveys and Monographs, 49 (1997).   Google Scholar [15] Y. Sugiyama and H. Kunii, Global existence and decay properties for a degenerate Keller-Segel model with a power factor in drift term,, J. Differential Equations, 227 (2006), 333.   Google Scholar [16] Y. Sugiyama and Y. Yahagi, Extinction, decay and blow-up for Keller-Segel systems of fast diffusion type,, J. Differential Equations, 250 (2011), 3047.   Google Scholar [17] Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity,, J. Differential Equations, 252 (2012), 692.   Google Scholar [18] M. Winkler, Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity,, Math. Nachr., 283 (2010), 1664.   Google Scholar [19] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model,, J. Differential Equations, 248 (2010), 2889.   Google Scholar [20] M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system,, J. Math. Pures Appl., 100 (2013), 748.   Google Scholar
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