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Existence of multiple spike stationary patterns in a chemotaxis model with weak saturation
1. | Department of Mathematics and Information Sciences, Tokyo Metropolitan University, 1-1 Minami-Ohsawa, Hachioji, Tokyo 192-0397, Japan |
References:
[1] |
P. Bates, E. N. Dancer and J. Shi, Multi-spike stationary solutions of the Cahn-Hilliard equation in higher-dimension and instability, Adv. Differential Equations, 4 (1999), 1-69. |
[2] |
P. Bates and J. Shi, Existence and instability of spike layer solutions to singular perturbation problems, J. Funct. Anal., 196 (2002), 211-264.
doi: 10.1016/S0022-1236(02)00013-7. |
[3] |
H. Berestycki and P.-L. Lions, Nonlinear scalar field equations I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-145.
doi: 10.1007/BF00250555. |
[4] |
H. Berestycki, T. Gallouët and O. Kavian, Nonlinear Euclidean scalar field equations in the plane, C. R. Acad. Sci. Paris Sér. I Math., 297 (1983), 307-310. |
[5] |
M. A. del Pino, Radially symmetric internal layers in a semilinear elliptic system, Trans. Amer. Math. Soc., 347 (1995), 4807-4837.
doi: 10.2307/2155064. |
[6] |
P. C. Fife, Semilinear elliptic boundary value problems with small parameters, Arch. Rational Mech. Anal., 52 (1973), 205-232.
doi: 10.1007/BF00247733. |
[7] |
D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165. |
[8] |
D. Iron, M. Ward and J. Wei, The stability of spike solutions to the one-dimensional Gierer-Meinhardt model, Phys. D, 150 (2001), 25-62.
doi: 10.1016/S0167-2789(00)00206-2. |
[9] |
T. Kolokolnikov, W. Sun, M. Ward and J. Wei, The stability of a stripe for the Gierer-Meinhardt model and the effect of saturation, SIAM J. Appl. Dyn. Syst., 5 (2006), 313-363.
doi: 10.1137/050635080. |
[10] |
K. Kurata and K. Morimoto, Construction and asymptotic behavior of multi-peak solutions to the Gierer-Meinhardt system with saturation, Commun. Pure Appl. Anal., 7 (2008), 1443-1482.
doi: 10.3934/cpaa.2008.7.1443. |
[11] |
M. K. Kwong and Y. Li, Uniqueness of radial solutions of semilinear elliptic equations, Trans. Amer. Math. Soc., 333 (1992), 339-363.
doi: 10.2307/2154113. |
[12] |
M. K. Kwong and L. Q. Zhang, Uniqueness of the positive solutions of $\Delta u+f(u)=0$ in an annulus, Differential Integral Equations, 4 (1991), 583-599. |
[13] |
H. A. Levine and B. D. Sleeman, A system of reaction diffusion equation arising in the theory of reinforced random walks, SIAM J. Appl. Math., 57 (1997), 683-730.
doi: 10.1137/S0036139995291106. |
[14] |
W.-M. Ni and I. Takagi, Point condensation generated by a reaction-diffusion system in axially symmetric domains, Japan J. Indust. Appl. Math., 12 (1995), 327-365.
doi: 10.1007/BF03167294. |
[15] |
W.-M. Ni, Qualitative properties of solutions to elliptic problems, Stationary Partial Differential Equations, I, Handb. Differ. Equ., North-Holland, Amsterdam, (2004), 157-233.
doi: 10.1016/S1874-5733(04)80005-6. |
[16] |
H. G. Othmer and A. Stevens, Aggregation, blowup, and collapse: The ABCs of taxis in reinforced random walks, SIAM J. Appl. Math., 57 (1997), 1044-1081.
doi: 10.1137/S0036139995288976. |
[17] |
T. Ouyang and J. Shi, Exact multiplicity of positive solutions for a class of semilinear problems. II, J. Differential Equations, 158 (1999), 94-151.
doi: 10.1016/S0022-0396(99)80020-5. |
[18] |
X. Ren and J. Wei, Oval shaped droplet solutions in the saturation process of some pattern formation problems, SIAM J. Appl. Math., 70 (2009), 1120-1138.
doi: 10.1137/080742361. |
[19] |
K. Sakamoto, Internal layers in high-dimensional domains, Proc. Roy. Soc. Edinburgh Sect. A., 128 (1998), 359-401. |
[20] |
T. Senba and T. Suzuki, "Applied Analysis. Mathematical Methods in Natural Science," Imperial College Press, London, 2004. |
[21] |
B. D. Sleeman, M. J. Ward and J. Wei, The existence and stability of spike patterns in a chemotaxis model, SIAM J. Appl Math., 65 (2005), 790-817.
doi: 10.1137/S0036139902415117. |
[22] |
T. Suzuki, "Free Energy and Self-Interacting Particles," Progress in Nonlinear Differential Equations and their Applications, 62, Birkäuser Boston, Inc., Boston, MA, 2005. |
[23] |
J. Wei, "Existence and Stability of Spikes for the Gierer-Meinhardt System," Handbook of Differential Equations: Stationary Partial Differential Equations, V, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, (2008), 487-585.
doi: 10.1016/S1874-5733(08)80013-7. |
[24] |
J. Wei and M. Winter, On the two-dimensional Gierer-Meinhardt system with strong coupling, SIAM J. Math. Anal., 30 (1999), 1241-1263.
doi: 10.1137/S0036141098347237. |
[25] |
J. Wei and M. Winter, Spikes for the Gierer-Meinhardt system in two dimensions: The strong coupling case, J. Differential Equations, 178 (2002), 478-518.
doi: 10.1006/jdeq.2001.4019. |
[26] |
J. Wei and M. Winter, On the Gierer-Meinhardt system with saturation, Commun. Contemp. Math., 6 (2004), 259-277.
doi: 10.1142/S021919970400132X. |
[27] |
J. Wei and M. Winter, Existence, classification and stability analysis of multiple-peaked solutions for the Gierer-Meinhardt system in $R^1$, Methods Appl. Anal., 14 (2007), 119-163. |
[28] |
J. Wei and M. Winter, Stationary multiple spots for reaction-diffusion systems, J. Math. Biol., 57 (2008), 53-89.
doi: 10.1007/s00285-007-0146-y. |
[29] |
E. Zeidler, "Nonlinear Functional Analysis and its Applications. I, Fixed-Point Theorems," Translated from the German by Peter R. Wadsack, Springer-Verlag, New York, 1986. |
show all references
References:
[1] |
P. Bates, E. N. Dancer and J. Shi, Multi-spike stationary solutions of the Cahn-Hilliard equation in higher-dimension and instability, Adv. Differential Equations, 4 (1999), 1-69. |
[2] |
P. Bates and J. Shi, Existence and instability of spike layer solutions to singular perturbation problems, J. Funct. Anal., 196 (2002), 211-264.
doi: 10.1016/S0022-1236(02)00013-7. |
[3] |
H. Berestycki and P.-L. Lions, Nonlinear scalar field equations I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-145.
doi: 10.1007/BF00250555. |
[4] |
H. Berestycki, T. Gallouët and O. Kavian, Nonlinear Euclidean scalar field equations in the plane, C. R. Acad. Sci. Paris Sér. I Math., 297 (1983), 307-310. |
[5] |
M. A. del Pino, Radially symmetric internal layers in a semilinear elliptic system, Trans. Amer. Math. Soc., 347 (1995), 4807-4837.
doi: 10.2307/2155064. |
[6] |
P. C. Fife, Semilinear elliptic boundary value problems with small parameters, Arch. Rational Mech. Anal., 52 (1973), 205-232.
doi: 10.1007/BF00247733. |
[7] |
D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165. |
[8] |
D. Iron, M. Ward and J. Wei, The stability of spike solutions to the one-dimensional Gierer-Meinhardt model, Phys. D, 150 (2001), 25-62.
doi: 10.1016/S0167-2789(00)00206-2. |
[9] |
T. Kolokolnikov, W. Sun, M. Ward and J. Wei, The stability of a stripe for the Gierer-Meinhardt model and the effect of saturation, SIAM J. Appl. Dyn. Syst., 5 (2006), 313-363.
doi: 10.1137/050635080. |
[10] |
K. Kurata and K. Morimoto, Construction and asymptotic behavior of multi-peak solutions to the Gierer-Meinhardt system with saturation, Commun. Pure Appl. Anal., 7 (2008), 1443-1482.
doi: 10.3934/cpaa.2008.7.1443. |
[11] |
M. K. Kwong and Y. Li, Uniqueness of radial solutions of semilinear elliptic equations, Trans. Amer. Math. Soc., 333 (1992), 339-363.
doi: 10.2307/2154113. |
[12] |
M. K. Kwong and L. Q. Zhang, Uniqueness of the positive solutions of $\Delta u+f(u)=0$ in an annulus, Differential Integral Equations, 4 (1991), 583-599. |
[13] |
H. A. Levine and B. D. Sleeman, A system of reaction diffusion equation arising in the theory of reinforced random walks, SIAM J. Appl. Math., 57 (1997), 683-730.
doi: 10.1137/S0036139995291106. |
[14] |
W.-M. Ni and I. Takagi, Point condensation generated by a reaction-diffusion system in axially symmetric domains, Japan J. Indust. Appl. Math., 12 (1995), 327-365.
doi: 10.1007/BF03167294. |
[15] |
W.-M. Ni, Qualitative properties of solutions to elliptic problems, Stationary Partial Differential Equations, I, Handb. Differ. Equ., North-Holland, Amsterdam, (2004), 157-233.
doi: 10.1016/S1874-5733(04)80005-6. |
[16] |
H. G. Othmer and A. Stevens, Aggregation, blowup, and collapse: The ABCs of taxis in reinforced random walks, SIAM J. Appl. Math., 57 (1997), 1044-1081.
doi: 10.1137/S0036139995288976. |
[17] |
T. Ouyang and J. Shi, Exact multiplicity of positive solutions for a class of semilinear problems. II, J. Differential Equations, 158 (1999), 94-151.
doi: 10.1016/S0022-0396(99)80020-5. |
[18] |
X. Ren and J. Wei, Oval shaped droplet solutions in the saturation process of some pattern formation problems, SIAM J. Appl. Math., 70 (2009), 1120-1138.
doi: 10.1137/080742361. |
[19] |
K. Sakamoto, Internal layers in high-dimensional domains, Proc. Roy. Soc. Edinburgh Sect. A., 128 (1998), 359-401. |
[20] |
T. Senba and T. Suzuki, "Applied Analysis. Mathematical Methods in Natural Science," Imperial College Press, London, 2004. |
[21] |
B. D. Sleeman, M. J. Ward and J. Wei, The existence and stability of spike patterns in a chemotaxis model, SIAM J. Appl Math., 65 (2005), 790-817.
doi: 10.1137/S0036139902415117. |
[22] |
T. Suzuki, "Free Energy and Self-Interacting Particles," Progress in Nonlinear Differential Equations and their Applications, 62, Birkäuser Boston, Inc., Boston, MA, 2005. |
[23] |
J. Wei, "Existence and Stability of Spikes for the Gierer-Meinhardt System," Handbook of Differential Equations: Stationary Partial Differential Equations, V, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, (2008), 487-585.
doi: 10.1016/S1874-5733(08)80013-7. |
[24] |
J. Wei and M. Winter, On the two-dimensional Gierer-Meinhardt system with strong coupling, SIAM J. Math. Anal., 30 (1999), 1241-1263.
doi: 10.1137/S0036141098347237. |
[25] |
J. Wei and M. Winter, Spikes for the Gierer-Meinhardt system in two dimensions: The strong coupling case, J. Differential Equations, 178 (2002), 478-518.
doi: 10.1006/jdeq.2001.4019. |
[26] |
J. Wei and M. Winter, On the Gierer-Meinhardt system with saturation, Commun. Contemp. Math., 6 (2004), 259-277.
doi: 10.1142/S021919970400132X. |
[27] |
J. Wei and M. Winter, Existence, classification and stability analysis of multiple-peaked solutions for the Gierer-Meinhardt system in $R^1$, Methods Appl. Anal., 14 (2007), 119-163. |
[28] |
J. Wei and M. Winter, Stationary multiple spots for reaction-diffusion systems, J. Math. Biol., 57 (2008), 53-89.
doi: 10.1007/s00285-007-0146-y. |
[29] |
E. Zeidler, "Nonlinear Functional Analysis and its Applications. I, Fixed-Point Theorems," Translated from the German by Peter R. Wadsack, Springer-Verlag, New York, 1986. |
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