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

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

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

Sachiko Ishida ^1, and Tomomi Yokota ^2,,

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<m+\frac{2}{N} $

. 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.

Keywords: Quasilinear degenerate Keller-Segel systems, boundedness.

Mathematics Subject Classification: Primary: 35K51; Secondary: 35B35.

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 L^p 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 L^p-L^q 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 L^p(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 L^p-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 L^p 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 L^p-L^q 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 L^p(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 L^p-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.

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