# American Institute of Mathematical Sciences

February  2020, 13(2): 211-232. doi: 10.3934/dcdss.2020012

## Boundedness in a quasilinear fully parabolic Keller-Segel system via maximal Sobolev regularity

 1 Department of Mathematics and Informatics, Graduate School of Science, Chiba University, 1-33, Yayoi-cho, Inage, Chiba 263-8522, Japan 2 Department of Mathematics, Tokyo University of Science, 1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan

* Corresponding author: Tomomi Yokota

Received  May 2017 Revised  October 2017 Published  January 2019

Fund Project: The first and second authors are supported by Grant-in-Aid for Young Scientists Research (B) (No. 15K17578) and Scientific Research (C) (No. 16K05182), JSPS, respectively.

This paper deals with the quasilinear Keller-Segel system
 \begin{align*} \begin{cases} u_t = \nabla\cdot(D(u)\nabla u)-\nabla\cdot(S(u)\nabla v), &x \in \Omega, \ t>0, \\ \ v_t = \Delta v - v +u, &x \in \Omega, \ t>0 \end{cases} \end{align*}
in
 $\Omega = \mathbb{R}^N$
or in a smoothly bounded domain
 $\Omega\subset \mathbb{R}^N$
, with nonnegative initial data
 $u_0\in L^1(\Omega) \cap L^\infty(\Omega)$
, and
 $v_0\in L^1(\Omega) \cap W^{1, \infty}(\Omega)$
; in the case that
 $\Omega$
is bounded, it is supplemented with homogeneous Neumann boundary condition. The diffusivity
 $D(u)$
and the sensitivity
 $S(u)$
are assumed to fulfill
 $D(u)\ge u^{m-1}\ (m\geq1)$
and
 $S(u)\leq u^{q-1}\ (q\geq 2)$
, respectively. This paper derives uniform-in-time boundedness of nonnegative solutions to the system when
 $q . In the case $ \Omega = \mathbb{R}^N $the result says boundedness which was not attained in a previous paper (J. Differential Equations 2012; 252:1421-1440). The proof is based on the maximal Sobolev regularity for the second equation. This also simplifies a previous proof given by Tao-Winkler (J. Differential Equations 2012; 252:692-715) in the case of bounded domains. Citation: Sachiko Ishida, Tomomi Yokota. Boundedness in a quasilinear fully parabolic Keller-Segel system via maximal Sobolev regularity. Discrete and Continuous Dynamical Systems - S, 2020, 13 (2) : 211-232. doi: 10.3934/dcdss.2020012 ##### References:  [1] H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I. Abstract Linear Theory, Monographs in Mathematics, 89, Birkhäuser Boston, 1995. doi: 10.1007/978-3-0348-9221-6. [2] N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763. doi: 10.1142/S021820251550044X. [3] X. Cao, Boundedness in a three-dimensional chemotaxis-haptotaxis model, Z. Angew. Math. Phys., 67 (2016), Art. 11, 13 pp. doi: 10.1007/s00033-015-0601-3. [4] P. Cannarsa and V. Vespri, On maximal Lp regularity for the abstract Cauchy problem, Boll. Un. Mat. Ital. B (6), 5 (1986), 165-175. [5] T. Ciéslak 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. doi: 10.1016/j.jde.2012.01.045. [6] K. Fujie, S. Ishida, A. Ito and T. Yokota, Large time behavior in a chemotaxis model with nonlinear general diffusion for tumor invasion, Funkcial. Ekvac., 61 (2018), 37-80. [7] M. Hieber and J. Prüss, Heat kernels and maximal Lp-Lq estimates for parabolic evolution equations, Comm. Partial Differential Equations, 22 (1997), 1647-1669. doi: 10.1080/03605309708821314. [8] T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3. [9] S. Ishida, An iterative approach to L∞-boundedness in quasilinear Keller-Segel systems, Discrete Contin. Dyn. Syst., 2015, Suppl., 635-643. doi: 10.3934/proc.2015.0635. [10] S. Ishida, Y. Maeda and T. Yokota, Gradient estimate for solutions to quasilinear non-degenerate Keller-Segel systems on$\mathbb{R}^N$, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2537-2568. doi: 10.3934/dcdsb.2013.18.2537. [11] S. Ishida, T. Ono and T. Yokota, Possibility of the existence of blow-up solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type, Math. Methods Appl. Sci., 36 (2013), 745-760. doi: 10.1002/mma.2622. [12] 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), 2993-3010. doi: 10.1016/j.jde.2014.01.028. [13] S. Ishida and T. Yokota, Global existence of weak solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type, J. Differential Equations, 252 (2012), 1421-1440. doi: 10.1016/j.jde.2011.02.012. [14] 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. doi: 10.1016/j.jde.2011.08.047. [15] S. Ishida and T. Yokota, Remaks on the global existence of weak solutions to quasilinear degenerate Keller-Segel systems, Discrete Contin. Dyn. Syst., 2013 (2013), 345-354. doi: 10.3934/proc.2013.2013.345. [16] 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. doi: 10.3934/dcdsb.2013.18.2569. [17] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. [18] S. Kim and K.-A. Lee, Hölder regularity and uniqueness theorem on weak solutions to the degenerate Keller-Segel system, Nonlinear Anal., 138 (2016), 229-252. doi: 10.1016/j.na.2015.11.024. [19] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, R. I., 1968. [20] M. Miura and Y. Sugiyama, On uniqueness theorem on weak solutions to the parabolic-parabolic Keller-Segel system of degenerate and singular types, J. Differential Equations, 257 (2014), 4064-4086. doi: 10.1016/j.jde.2014.08.001. [21] K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543. [22] T. Senba and T. Suzuki, A quasi-linear parabolic system of chemotaxis, Abstr. Appl. Anal., 2006 (2006), Art. ID 23061, 21 pp. doi: 10.1155/AAA/2006/23061. [23] J. Simon, Compact sets in the space Lp(0, T; B), Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360. [24] 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. doi: 10.1016/j.jde.2006.03.003. [25] Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715. doi: 10.1016/j.jde.2011.08.019. [26] P. Weidemaier, Maximal regularity for parabolic equations with inhomogeneous boundary conditions in Sobolev spaces with mixed Lp-norm, Electron. Res. Announc. Amer. Math. Soc., 8 (2002), 47-51. doi: 10.1090/S1079-6762-02-00104-X. [27] M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24. doi: 10.1002/mma.1146. [28] M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767. doi: 10.1016/j.matpur.2013.01.020. show all references ##### References:  [1] H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I. Abstract Linear Theory, Monographs in Mathematics, 89, Birkhäuser Boston, 1995. doi: 10.1007/978-3-0348-9221-6. [2] N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763. doi: 10.1142/S021820251550044X. [3] X. Cao, Boundedness in a three-dimensional chemotaxis-haptotaxis model, Z. Angew. Math. Phys., 67 (2016), Art. 11, 13 pp. doi: 10.1007/s00033-015-0601-3. [4] P. Cannarsa and V. Vespri, On maximal Lp regularity for the abstract Cauchy problem, Boll. Un. Mat. Ital. B (6), 5 (1986), 165-175. [5] T. Ciéslak 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. doi: 10.1016/j.jde.2012.01.045. [6] K. Fujie, S. Ishida, A. Ito and T. Yokota, Large time behavior in a chemotaxis model with nonlinear general diffusion for tumor invasion, Funkcial. Ekvac., 61 (2018), 37-80. [7] M. Hieber and J. Prüss, Heat kernels and maximal Lp-Lq estimates for parabolic evolution equations, Comm. Partial Differential Equations, 22 (1997), 1647-1669. doi: 10.1080/03605309708821314. [8] T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3. [9] S. Ishida, An iterative approach to L∞-boundedness in quasilinear Keller-Segel systems, Discrete Contin. Dyn. Syst., 2015, Suppl., 635-643. doi: 10.3934/proc.2015.0635. [10] S. Ishida, Y. Maeda and T. Yokota, Gradient estimate for solutions to quasilinear non-degenerate Keller-Segel systems on$\mathbb{R}^N$, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2537-2568. doi: 10.3934/dcdsb.2013.18.2537. [11] S. Ishida, T. Ono and T. Yokota, Possibility of the existence of blow-up solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type, Math. Methods Appl. Sci., 36 (2013), 745-760. doi: 10.1002/mma.2622. [12] 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), 2993-3010. doi: 10.1016/j.jde.2014.01.028. [13] S. Ishida and T. Yokota, Global existence of weak solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type, J. Differential Equations, 252 (2012), 1421-1440. doi: 10.1016/j.jde.2011.02.012. [14] 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. doi: 10.1016/j.jde.2011.08.047. [15] S. Ishida and T. Yokota, Remaks on the global existence of weak solutions to quasilinear degenerate Keller-Segel systems, Discrete Contin. Dyn. Syst., 2013 (2013), 345-354. doi: 10.3934/proc.2013.2013.345. [16] 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. doi: 10.3934/dcdsb.2013.18.2569. [17] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. [18] S. Kim and K.-A. Lee, Hölder regularity and uniqueness theorem on weak solutions to the degenerate Keller-Segel system, Nonlinear Anal., 138 (2016), 229-252. doi: 10.1016/j.na.2015.11.024. [19] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, R. I., 1968. [20] M. Miura and Y. Sugiyama, On uniqueness theorem on weak solutions to the parabolic-parabolic Keller-Segel system of degenerate and singular types, J. Differential Equations, 257 (2014), 4064-4086. doi: 10.1016/j.jde.2014.08.001. [21] K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543. [22] T. Senba and T. Suzuki, A quasi-linear parabolic system of chemotaxis, Abstr. Appl. Anal., 2006 (2006), Art. ID 23061, 21 pp. doi: 10.1155/AAA/2006/23061. [23] J. Simon, Compact sets in the space Lp(0, T; B), Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360. [24] 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. doi: 10.1016/j.jde.2006.03.003. [25] Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715. doi: 10.1016/j.jde.2011.08.019. [26] P. Weidemaier, Maximal regularity for parabolic equations with inhomogeneous boundary conditions in Sobolev spaces with mixed Lp-norm, Electron. Res. Announc. Amer. Math. Soc., 8 (2002), 47-51. doi: 10.1090/S1079-6762-02-00104-X. [27] M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24. doi: 10.1002/mma.1146. [28] M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767. doi: 10.1016/j.matpur.2013.01.020.  [1] Sachiko Ishida. An iterative approach to$L^\infty$-boundedness in quasilinear Keller-Segel systems. Conference Publications, 2015, 2015 (special) : 635-643. doi: 10.3934/proc.2015.0635 [2] Xie Li, Zhaoyin Xiang. Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3503-3531. doi: 10.3934/dcds.2015.35.3503 [3] Sachiko Ishida, Tomomi Yokota. Remarks on the global existence of weak solutions to quasilinear degenerate Keller-Segel systems. Conference Publications, 2013, 2013 (special) : 345-354. doi: 10.3934/proc.2013.2013.345 [4] Sachiko Ishida, Tomomi Yokota. Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type. Discrete and Continuous Dynamical Systems - B, 2013, 18 (10) : 2569-2596. doi: 10.3934/dcdsb.2013.18.2569 [5] Sachiko Ishida.$L^\infty$-decay property for quasilinear degenerate parabolic-elliptic Keller-Segel systems. Conference Publications, 2013, 2013 (special) : 335-344. doi: 10.3934/proc.2013.2013.335 [6] Sachiko Ishida, Yusuke Maeda, Tomomi Yokota. Gradient estimate for solutions to quasilinear non-degenerate Keller-Segel systems on$\mathbb{R}^N$. Discrete and Continuous Dynamical Systems - B, 2013, 18 (10) : 2537-2568. doi: 10.3934/dcdsb.2013.18.2537 [7] Kentarou Fujie, Chihiro Nishiyama, Tomomi Yokota. Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with the sensitivity$v^{-1}S(u)$. Conference Publications, 2015, 2015 (special) : 464-472. doi: 10.3934/proc.2015.0464 [8] Wenting Cong, Jian-Guo Liu. Uniform$L^{∞}$boundedness for a degenerate parabolic-parabolic Keller-Segel model. Discrete and Continuous Dynamical Systems - B, 2017, 22 (2) : 307-338. doi: 10.3934/dcdsb.2017015 [9] Wenting Cong, Jian-Guo Liu. A degenerate$p$-Laplacian Keller-Segel model. Kinetic and Related Models, 2016, 9 (4) : 687-714. doi: 10.3934/krm.2016012 [10] Mengyao Ding, Xiangdong Zhao.$ L^\sigma $-measure criteria for boundedness in a quasilinear parabolic-parabolic Keller-Segel system with supercritical sensitivity. 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