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Computer-assisted equilibrium validation for the diblock copolymer model
Dynamics of spike in a Keller-Segel's minimal chemotaxis model
1. | School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China |
2. | Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA |
3. | Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China |
4. | Center for Financial Engineering, Soochow University, Suzhou, Jiangsu 215006, China |
$τ u_t=(u_x-kuv_x)_x, \ \ \ \ v_t=v_{xx}-v+u$ |
$\tau\geqslant 0$ |
$k\gg 1$ |
$u$ |
$u$ |
$v$ |
$v$ |
$\tau$ |
References:
[1] |
N. Alikakos, P. W. Bates and G. Fusco,
Slow motion for the Cahn-Hilliard equation in one space dimension, J. Differ. Eqns., 90 (1991), 81-135.
doi: 10.1016/0022-0396(91)90163-4. |
[2] |
P. W. Bates and J. Xun,
Metastable patterns for the Cahn-Hilliard equation, Part Ⅰ, J. Differ. Eqns., 111 (1994), 421-457.
doi: 10.1006/jdeq.1994.1089. |
[3] |
P. W. Bates and J. Xun,
Metastable patterns for the Cahn-Hilliard equation, Part Ⅱ, J. Differ. Eqns., 117 (1995), 165-216.
doi: 10.1006/jdeq.1995.1052. |
[4] |
P. Biler,
Local and global solvability of some parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743.
|
[5] |
L. Bronsard and D. Hilhorst,
On the slow dynamics for the Cahn-Hilliard equation in one space dimension, Proc. Roy. Soc. Lond., 439 (1992), 669-682.
doi: 10.1098/rspa.1992.0176. |
[6] |
J. Carr and R. Pego,
Metastable patterns in solutions of $u_t =\varepsilon ^2 u_{xx}- f(u)$, Comm. Pure Appl. Math., 42 (1989), 523-576.
doi: 10.1002/cpa.3160420502. |
[7] |
X. Chen,
Generation, propagation, and annihilation of metastable patterns, J. Differ. Eqns., 206 (2004), 399-437.
doi: 10.1016/j.jde.2004.05.017. |
[8] |
X. Chen, J. Hao, X. Wang, Y. Wu and Y. Zhang,
Stability of spiky solution of the Keller-Segel's minimal chemotaxis model, J. Differ. Eqns., 257 (2014), 3102-3134.
doi: 10.1016/j.jde.2014.06.008. |
[9] |
X. Chen and M. Kowalczyk,
Dynamics of an interior sipke in the Gierer-Meinhardt system, Siam J. Math. Anal., 33 (2001), 172-193.
doi: 10.1137/S0036141099364954. |
[10] |
X. Chen and M. Kowalczyk,
Slow dynamics of interior spikes in the Shadow Gierer-Meinhardt system, Adv. Differ. Eqns., 6 (2001), 847-872.
|
[11] |
S. Childress, Chemotactic collapse in two dimensions, in Lecture Notes in Biomath. , 55, Springer, (1984), 61-66. |
[12] |
S. Childress and J. Perkus,
Nonlinear aspects of chemotaxis, Math. Bios., 56 (1981), 217-237.
doi: 10.1016/0025-5564(81)90055-9. |
[13] |
P. C. Fife and L. Hsiao,
The generation and propagation of internal layers, Nonlinear Anal., 12 (1988), 19-41.
doi: 10.1016/0362-546X(88)90010-7. |
[14] |
P. C. Fife and J. B. McLeod,
The approach of solutions of nonlinear diffusion equations to traveling front solutions, Arch. Rational Mech. Anal., 65 (1977), 335-361.
doi: 10.1007/BF00250432. |
[15] |
G. Fusco, A geometric approach to the dynamics of $u_t =\varepsilon ^2 u_{xx} +f(u)$ for small $\varepsilon $, in Problems Involving Change of Type, Springer, 359 (1990), 53-73. |
[16] |
G. Fusco and J. K. Hale,
Slow motion manifolds, dormant instability and singular perturbations, J. Dyn. Diff. Eqns., 1 (1989), 75-94.
doi: 10.1007/BF01048791. |
[17] |
H. Gajewski, K. Zacharias and Dr. Konrad Gröger,
Global behavior of a reaction-diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77-114.
doi: 10.1002/mana.19981950106. |
[18] |
M. Herrero and J. Velázquez,
Singularity patterns in a chemotaxis model, Math. Ann., 306 (1996), 583-623.
doi: 10.1007/BF01445268. |
[19] |
M. Herrero and J. Velázquez,
Chemotaxis collapse for the Keller-Segel model, J. Math. Biol., 35 (1996), 177-194.
doi: 10.1007/s002850050049. |
[20] |
T. Hillen and K. J. Painter,
Global existence far a parabolic chemotaxis model with prevention of overcrowding, Adv. Appl. Math., 26 (2001), 280-301.
doi: 10.1006/aama.2001.0721. |
[21] |
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. |
[22] |
T. Hillen and A. Potapov,
The one-dimensional chemotaxis model: Global existence and asymptotic profile, Math. Meth. Appl. Sci., 27 (2004), 1783-1801.
doi: 10.1002/mma.569. |
[23] |
D. Horstmann,
From 1970 until now: The Keller-Segal model in chemotaxis and its consequences, Ⅰ, Jahresber. DMV, 105 (2003), 103-165.
|
[24] |
D. Horstmann,
From 1970 until now: The Keller-Segal model in chemotaxis and its consequences, Ⅱ, Jahresber. DMV, 106 (2004), 51-69.
|
[25] |
W. Jäger and S. Luckhaus,
On explosions of solutions to a system of partial differential equations modellingchemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824.
doi: 10.2307/2153966. |
[26] |
K. Kang, T. Kolokolnikov and M. J. Ward,
The stability and dynamics of a spike in the one-dimensional Keller-Segel model, IMA J. Appl. Math., 72 (2007), 140-162.
doi: 10.1093/imamat/hxl028. |
[27] |
E. Keller and L. Segel,
Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[28] |
C.-S. Lin, W.-M. Ni and I. Takagi,
Large amplitude stationary solutions to a chemotaxis system, J. Differ. Eqns., 72 (1988), 1-27.
doi: 10.1016/0022-0396(88)90147-7. |
[29] |
T. Nagai,
Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.
|
[30] |
T. Nagai, T. Senba and K. Yoshida,
Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433.
|
[31] |
T. Nagai,
Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in twodimensional domains, J. Inequal. Appl., 6 (2001), 37-55.
doi: 10.1155/S1025583401000042. |
[32] |
W.-M. Ni and I. Takagi,
Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J., 70 (1993), 247-281.
doi: 10.1215/S0012-7094-93-07004-4. |
[33] |
K. Osaki and A. Yagi,
Finite dimensional attractors for one dimensional Keller-Segel equations, Funkcial Ekvac., 44 (2001), 441-469.
|
[34] |
C. S. Patlak,
Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338.
doi: 10.1007/BF02476407. |
[35] |
T. Senba and T. Suzuki,
Parabolic system of chemotaxis: Blowup in a finite and the infinite time, Methods Appl. Anal., 8 (2001), 349-367.
doi: 10.4310/MAA.2001.v8.n2.a9. |
[36] |
X. Sun and M. J. Ward,
Dynamics and coarsening of interfaces for the viscous Cahn-Hilliard equation in one spatial dimension, Stud. Appl. Math., 105 (2000), 203-234.
doi: 10.1111/1467-9590.00149. |
[37] |
X. Wang and Q. Xu,
Spiky and transition layer steady states of chemotaxis systems via global bifurcation and Helly compactness theorem, J. Math. Biol., 66 (2013), 1241-1266.
doi: 10.1007/s00285-012-0533-x. |
[38] |
Y. Zhang, X. Chen, J. Hao, X. Lai and C. Qin,
An eigenvalue problem arising from spiky steady states of a minimal chemotaxis model, J. Math. Anal. Appl., 420 (2014), 684-704.
doi: 10.1016/j.jmaa.2014.06.005. |
show all references
References:
[1] |
N. Alikakos, P. W. Bates and G. Fusco,
Slow motion for the Cahn-Hilliard equation in one space dimension, J. Differ. Eqns., 90 (1991), 81-135.
doi: 10.1016/0022-0396(91)90163-4. |
[2] |
P. W. Bates and J. Xun,
Metastable patterns for the Cahn-Hilliard equation, Part Ⅰ, J. Differ. Eqns., 111 (1994), 421-457.
doi: 10.1006/jdeq.1994.1089. |
[3] |
P. W. Bates and J. Xun,
Metastable patterns for the Cahn-Hilliard equation, Part Ⅱ, J. Differ. Eqns., 117 (1995), 165-216.
doi: 10.1006/jdeq.1995.1052. |
[4] |
P. Biler,
Local and global solvability of some parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743.
|
[5] |
L. Bronsard and D. Hilhorst,
On the slow dynamics for the Cahn-Hilliard equation in one space dimension, Proc. Roy. Soc. Lond., 439 (1992), 669-682.
doi: 10.1098/rspa.1992.0176. |
[6] |
J. Carr and R. Pego,
Metastable patterns in solutions of $u_t =\varepsilon ^2 u_{xx}- f(u)$, Comm. Pure Appl. Math., 42 (1989), 523-576.
doi: 10.1002/cpa.3160420502. |
[7] |
X. Chen,
Generation, propagation, and annihilation of metastable patterns, J. Differ. Eqns., 206 (2004), 399-437.
doi: 10.1016/j.jde.2004.05.017. |
[8] |
X. Chen, J. Hao, X. Wang, Y. Wu and Y. Zhang,
Stability of spiky solution of the Keller-Segel's minimal chemotaxis model, J. Differ. Eqns., 257 (2014), 3102-3134.
doi: 10.1016/j.jde.2014.06.008. |
[9] |
X. Chen and M. Kowalczyk,
Dynamics of an interior sipke in the Gierer-Meinhardt system, Siam J. Math. Anal., 33 (2001), 172-193.
doi: 10.1137/S0036141099364954. |
[10] |
X. Chen and M. Kowalczyk,
Slow dynamics of interior spikes in the Shadow Gierer-Meinhardt system, Adv. Differ. Eqns., 6 (2001), 847-872.
|
[11] |
S. Childress, Chemotactic collapse in two dimensions, in Lecture Notes in Biomath. , 55, Springer, (1984), 61-66. |
[12] |
S. Childress and J. Perkus,
Nonlinear aspects of chemotaxis, Math. Bios., 56 (1981), 217-237.
doi: 10.1016/0025-5564(81)90055-9. |
[13] |
P. C. Fife and L. Hsiao,
The generation and propagation of internal layers, Nonlinear Anal., 12 (1988), 19-41.
doi: 10.1016/0362-546X(88)90010-7. |
[14] |
P. C. Fife and J. B. McLeod,
The approach of solutions of nonlinear diffusion equations to traveling front solutions, Arch. Rational Mech. Anal., 65 (1977), 335-361.
doi: 10.1007/BF00250432. |
[15] |
G. Fusco, A geometric approach to the dynamics of $u_t =\varepsilon ^2 u_{xx} +f(u)$ for small $\varepsilon $, in Problems Involving Change of Type, Springer, 359 (1990), 53-73. |
[16] |
G. Fusco and J. K. Hale,
Slow motion manifolds, dormant instability and singular perturbations, J. Dyn. Diff. Eqns., 1 (1989), 75-94.
doi: 10.1007/BF01048791. |
[17] |
H. Gajewski, K. Zacharias and Dr. Konrad Gröger,
Global behavior of a reaction-diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77-114.
doi: 10.1002/mana.19981950106. |
[18] |
M. Herrero and J. Velázquez,
Singularity patterns in a chemotaxis model, Math. Ann., 306 (1996), 583-623.
doi: 10.1007/BF01445268. |
[19] |
M. Herrero and J. Velázquez,
Chemotaxis collapse for the Keller-Segel model, J. Math. Biol., 35 (1996), 177-194.
doi: 10.1007/s002850050049. |
[20] |
T. Hillen and K. J. Painter,
Global existence far a parabolic chemotaxis model with prevention of overcrowding, Adv. Appl. Math., 26 (2001), 280-301.
doi: 10.1006/aama.2001.0721. |
[21] |
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. |
[22] |
T. Hillen and A. Potapov,
The one-dimensional chemotaxis model: Global existence and asymptotic profile, Math. Meth. Appl. Sci., 27 (2004), 1783-1801.
doi: 10.1002/mma.569. |
[23] |
D. Horstmann,
From 1970 until now: The Keller-Segal model in chemotaxis and its consequences, Ⅰ, Jahresber. DMV, 105 (2003), 103-165.
|
[24] |
D. Horstmann,
From 1970 until now: The Keller-Segal model in chemotaxis and its consequences, Ⅱ, Jahresber. DMV, 106 (2004), 51-69.
|
[25] |
W. Jäger and S. Luckhaus,
On explosions of solutions to a system of partial differential equations modellingchemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824.
doi: 10.2307/2153966. |
[26] |
K. Kang, T. Kolokolnikov and M. J. Ward,
The stability and dynamics of a spike in the one-dimensional Keller-Segel model, IMA J. Appl. Math., 72 (2007), 140-162.
doi: 10.1093/imamat/hxl028. |
[27] |
E. Keller and L. Segel,
Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[28] |
C.-S. Lin, W.-M. Ni and I. Takagi,
Large amplitude stationary solutions to a chemotaxis system, J. Differ. Eqns., 72 (1988), 1-27.
doi: 10.1016/0022-0396(88)90147-7. |
[29] |
T. Nagai,
Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.
|
[30] |
T. Nagai, T. Senba and K. Yoshida,
Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433.
|
[31] |
T. Nagai,
Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in twodimensional domains, J. Inequal. Appl., 6 (2001), 37-55.
doi: 10.1155/S1025583401000042. |
[32] |
W.-M. Ni and I. Takagi,
Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J., 70 (1993), 247-281.
doi: 10.1215/S0012-7094-93-07004-4. |
[33] |
K. Osaki and A. Yagi,
Finite dimensional attractors for one dimensional Keller-Segel equations, Funkcial Ekvac., 44 (2001), 441-469.
|
[34] |
C. S. Patlak,
Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338.
doi: 10.1007/BF02476407. |
[35] |
T. Senba and T. Suzuki,
Parabolic system of chemotaxis: Blowup in a finite and the infinite time, Methods Appl. Anal., 8 (2001), 349-367.
doi: 10.4310/MAA.2001.v8.n2.a9. |
[36] |
X. Sun and M. J. Ward,
Dynamics and coarsening of interfaces for the viscous Cahn-Hilliard equation in one spatial dimension, Stud. Appl. Math., 105 (2000), 203-234.
doi: 10.1111/1467-9590.00149. |
[37] |
X. Wang and Q. Xu,
Spiky and transition layer steady states of chemotaxis systems via global bifurcation and Helly compactness theorem, J. Math. Biol., 66 (2013), 1241-1266.
doi: 10.1007/s00285-012-0533-x. |
[38] |
Y. Zhang, X. Chen, J. Hao, X. Lai and C. Qin,
An eigenvalue problem arising from spiky steady states of a minimal chemotaxis model, J. Math. Anal. Appl., 420 (2014), 684-704.
doi: 10.1016/j.jmaa.2014.06.005. |


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