# American Institute of Mathematical Sciences

2015, 2015(special): 464-472. doi: 10.3934/proc.2015.0464

## Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with the sensitivity $v^{-1}S(u)$

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

Received  July 2014 Revised  November 2014 Published  November 2015

This paper is concerned with global existence and boundedness of classical solutions to the quasilinear fully parabolic Keller-Segel system $u_t = \nabla \cdot(D(u)\nabla u) -\nabla \cdot (v^{-1}S(u)\nabla v)$, $v_t= \Delta v-v+u$. In [7,4], global existence and boundedness were established in the system without $v^{-1}$. In this paper the signal-dependent sensitivity $v^{-1}$ is taken into account via the Weber-Fechner law. A uniform-in-time estimate for $v$ obtained in [2] defeats the singularity of $v^{-1}$.
Citation: Kentarou Fujie, Chihiro Nishiyama, Tomomi Yokota. Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with the sensitivity $v^{-1}S(u)$. Conference Publications, 2015, 2015 (special) : 464-472. doi: 10.3934/proc.2015.0464
##### References:
 [1] T. Ciéslak, 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. [2] K. Fujie, Boundedness in a fully parabolic chemotaxis system with singular sensitivity,, J. Math. Anal. Appl., 424 (2015), 675. [3] T. Hillen, K. J. Painter, A user's guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183. [4] S. Ishida, K. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains,, J. Differential Equations, 256 (2014), 2993. [5] E. F. Keller, L. A. Segel, Initiation of slime mold aggregation viewed as an instability ,, J. Theor. Biol., 26 (1970), 399. [6] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type,, Amer. Math. Soc. Transl., (1968). [7] Y. Tao, M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity,, J. Differential Equations, 252 (2012), 692. [8] M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?,, Math. Methods Appl. Sci., 33 (2010), 12. [9] M. Winkler, Global solutions in a fully parabolic chemotaxis system with singular sensitivity,, Math. Methods Appl. Sci., 34 (2011), 176. [10] M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system,, J. Math. Pures Appl., 100 (2013), 748.

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##### References:
 [1] T. Ciéslak, 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. [2] K. Fujie, Boundedness in a fully parabolic chemotaxis system with singular sensitivity,, J. Math. Anal. Appl., 424 (2015), 675. [3] T. Hillen, K. J. Painter, A user's guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183. [4] S. Ishida, K. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains,, J. Differential Equations, 256 (2014), 2993. [5] E. F. Keller, L. A. Segel, Initiation of slime mold aggregation viewed as an instability ,, J. Theor. Biol., 26 (1970), 399. [6] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type,, Amer. Math. Soc. Transl., (1968). [7] Y. Tao, M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity,, J. Differential Equations, 252 (2012), 692. [8] M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?,, Math. Methods Appl. Sci., 33 (2010), 12. [9] M. Winkler, Global solutions in a fully parabolic chemotaxis system with singular sensitivity,, Math. Methods Appl. Sci., 34 (2011), 176. [10] M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system,, J. Math. Pures Appl., 100 (2013), 748.
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