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Infinite time blow-up of many solutions to a general quasilinear parabolic-elliptic Keller-Segel system
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Global asymptotic stability in a chemotaxis-growth model for tumor invasion
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 |
$ \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*} $ |
$ \Omega = \mathbb{R}^N $ |
$ \Omega\subset \mathbb{R}^N $ |
$ u_0\in L^1(\Omega) \cap L^\infty(\Omega) $ |
$ v_0\in L^1(\Omega) \cap W^{1, \infty}(\Omega) $ |
$ \Omega $ |
$ D(u) $ |
$ S(u) $ |
$ D(u)\ge u^{m-1}\ (m\geq1) $ |
$ S(u)\leq u^{q-1}\ (q\geq 2) $ |
$ q<m+\frac{2}{N} $ |
$ \Omega = \mathbb{R}^N $ |
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. |
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