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<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.
Citation: Sachiko Ishida, Tomomi Yokota. Boundedness in a quasilinear fully parabolic Keller-Segel system via maximal Sobolev regularity. Discrete & Continuous Dynamical Systems - S, 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. Google Scholar

[2]

N. BellomoA. BellouquidY. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[6]

K. FujieS. IshidaA. Ito and T. Yokota, Large time behavior in a chemotaxis model with nonlinear general diffusion for tumor invasion, Funkcial. Ekvac., 61 (2018), 37-80. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

S. IshidaY. 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. Google Scholar

[11]

S. IshidaT. 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. Google Scholar

[12]

S. IshidaK. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[17]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[23]

J. Simon, Compact sets in the space Lp(0, T; B), Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[2]

N. BellomoA. BellouquidY. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[6]

K. FujieS. IshidaA. Ito and T. Yokota, Large time behavior in a chemotaxis model with nonlinear general diffusion for tumor invasion, Funkcial. Ekvac., 61 (2018), 37-80. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

S. IshidaY. 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. Google Scholar

[11]

S. IshidaT. 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. Google Scholar

[12]

S. IshidaK. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[17]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[23]

J. Simon, Compact sets in the space Lp(0, T; B), Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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 & Continuous Dynamical Systems - A, 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 & 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 & 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 & 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 & Related Models, 2016, 9 (4) : 687-714. doi: 10.3934/krm.2016012

[10]

Miaoqing Tian, Sining Zheng. Global boundedness versus finite-time blow-up of solutions to a quasilinear fully parabolic Keller-Segel system of two species. Communications on Pure & Applied Analysis, 2016, 15 (1) : 243-260. doi: 10.3934/cpaa.2016.15.243

[11]

Mengyao Ding, Sining Zheng. $ L^γ$-measure criteria for boundedness in a quasilinear parabolic-elliptic Keller-Segel system with supercritical sensitivity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 2971-2988. doi: 10.3934/dcdsb.2018295

[12]

Mengyao Ding, Xiangdong Zhao. $ L^\sigma $-measure criteria for boundedness in a quasilinear parabolic-parabolic Keller-Segel system with supercritical sensitivity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5297-5315. doi: 10.3934/dcdsb.2019059

[13]

Hao Yu, Wei Wang, Sining Zheng. Boundedness of solutions to a fully parabolic Keller-Segel system with nonlinear sensitivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1635-1644. doi: 10.3934/dcdsb.2017078

[14]

Hao Yu, Wei Wang, Sining Zheng. Global boundedness of solutions to a Keller-Segel system with nonlinear sensitivity. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1317-1327. doi: 10.3934/dcdsb.2016.21.1317

[15]

Qi Wang, Jingyue Yang, Feng Yu. Boundedness in logistic Keller-Segel models with nonlinear diffusion and sensitivity functions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 5021-5036. doi: 10.3934/dcds.2017216

[16]

Jan Burczak, Rafael Granero-Belinchón. Boundedness and homogeneous asymptotics for a fractional logistic Keller-Segel equations. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 139-164. doi: 10.3934/dcdss.2020008

[17]

Kenneth H. Karlsen, Süleyman Ulusoy. On a hyperbolic Keller-Segel system with degenerate nonlinear fractional diffusion. Networks & Heterogeneous Media, 2016, 11 (1) : 181-201. doi: 10.3934/nhm.2016.11.181

[18]

Kentarou Fujie, Takasi Senba. Global existence and boundedness in a parabolic-elliptic Keller-Segel system with general sensitivity. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 81-102. doi: 10.3934/dcdsb.2016.21.81

[19]

Johannes Lankeit. Infinite time blow-up of many solutions to a general quasilinear parabolic-elliptic Keller-Segel system. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 233-255. doi: 10.3934/dcdss.2020013

[20]

Yoshifumi Mimura. Critical mass of degenerate Keller-Segel system with no-flux and Neumann boundary conditions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1603-1630. doi: 10.3934/dcds.2017066

2018 Impact Factor: 0.545

Article outline

[Back to Top]