Article Contents
Article Contents

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

• 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.
Mathematics Subject Classification: Primary: 35K51; Secondary: 35D30, 35Q32, 92C17.

 Citation:

•  [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), 5832-5851. [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), 135-146. [3] M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa Cl. Sci., (4) 24 (1997), 633-683. [4] T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. [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), 2537-2568. [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), 2469-2491. [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), 345-354. [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) 2569-2596. [9] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. [10] T. Nagai, Global existence and blowup of solutions to a chemotaxis system, Nonlinear Anal., 47 (2001), 777-787. [11] T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433. [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), 383-407. [13] K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469. [14] R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Mathematical Surveys and Monographs, 49 American Mathematical Society, Providence, RI, (1997). [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), 333-364. [16] Y. Sugiyama and Y. Yahagi, Extinction, decay and blow-up for Keller-Segel systems of fast diffusion type, J. Differential Equations, 250 (2011), 3047-3087. [17] Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715. [18] M. Winkler, Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity, Math. Nachr., 283 (2010), 1664-1673. [19] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905. [20] M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.
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