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Qualitative analysis and travelling wave solutions for the SI model with vertical transmission
Dynamics of a boundary spike for the shadow Gierer-Meinhardt system
1. | Faculty of Mathematics, Kyushu University, Fukuoka 819-0395, Japan |
2. | Graduate School of Advanced Mathematical Science, Meiji University, Kawasaki, 214-8571, Japan |
3. | Department of Mathematics, Tokyo Institute of Technology, O-Okayama, Meguro-ku, Tokyo 152-8551 |
References:
[1] |
N. D. Alikakos, P. W. Bates, X. Chen and G. Fusco, Mullins-Sekerka motion of small droplets on a fixed boundary, J. Geom. Anal., 10 (2000), 575-596. |
[2] |
P. W. Bates, K. Lu and C. Zeng, Approximately invariant manifolds and global dynamics of spike states, Invent. Math., 174 (2008), 355-433. |
[3] |
X. Chen and M. Kowalczyk, Slow dynamics of interior spikes in the shadow Gierer-Meinhardt system, Adv. Differential Equations, 6 (2001), 847-872. |
[4] |
M. del Pino, P. L. Felmer and M. Kowalczyk, Boundary spikes in the Gierer-Meinhardt system, Commun. Pure Appl. Anal., 1 (2002), 437-456.
doi: 10.3934/cpaa.2002.1.437. |
[5] |
M. del Pino, P. L. Felmer and J. Wei, On the role of mean curvature in some singularly perturbed Neumann problems, SIAM J. Math. Anal., 31 (1999), 63-79 (electronic).
doi: 10.1137/S0036141098332834. |
[6] |
M. del Pino, P. L. Felmer and J. Wei, On the role of distance function in some singular perturbation problems, Comm. Partial Differential Equations, 25 (2000), 155-177.
doi: 10.1080/03605300008821511. |
[7] |
S.-I. Ei, The motion of weakly interacting pulses in reaction-diffusion systems, J. D. D. E. 14 (2002), 85-137.
doi: 10.1023/A:1012980128575. |
[8] |
S.-I. Ei, Dynamics and their interaction of spikes on smoothly curved boundaries for reaction-diffusion systems in 2D, preprint. |
[9] |
L. C. Evans, "Partial Differential Equations,'' Graduate Studies in Mathematics, 19, (1998). |
[10] |
B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$, Mathematical analysis and applications, Part A, Adv. in Math. Suppl. Stud., 12 (1972), 30-39. Academic Press, (1981), 369-402. |
[11] |
A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetic, 12 (1972), 30-39.
doi: 10.1007/BF00289234. |
[12] |
C. Gui and J. Wei, On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems, Canad. J. Math., 52 (2000), 522-538.
doi: 10.4153/CJM-2000-024-x. |
[13] |
D. Iron and M. J. Ward, The dynamics of boundary spikes for a nonlocal reaction-diffusion model, European J. Appl. Math., 11 (2000), 491-514.
doi: 10.1017/S0956792500004253. |
[14] |
D. Iron, M. J. 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. |
[15] |
C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27.
doi: 10.1016/0022-0396(88)90147-7. |
[16] |
H. Meinhardt, "Models of Biological Pattern Formation,'' Academic Press, 1982. |
[17] |
Y. Miyamoto, Stability of a boundary spike layer for the Gierer-Meinhardt system, European J. Appl. Math., 16 (2005), 467-491.
doi: 10.1017/S0956792505006376. |
[18] |
Y. Miyamoto, On the shape of the stable patterns for activator-inhibitor systems in two-dimensional domains, Quarterly of Applied Mathematics, 65 (2007), 357-374. |
[19] |
W.-M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math., 44 (1991), 819-851.
doi: 10.1002/cpa.3160440705. |
[20] |
W.-M. Ni and I. Takagi, Locating the peaks of least energy solution to a semilinear Neumann problem, Duke Math. J., 70 (1993), 247-281.
doi: 10.1215/S0012-7094-93-07004-4. |
[21] |
W.-M. Ni, I. Takagi and E. Yanagida, Stability of least energy patterns of the shadow system for an activator-inhibitor model, Japan J. Indust. Appl. Math., 18, (2001), 259-272. |
[22] |
Y. Nishiura, Global structure of bifurcating solutions of some reaction-diffusion systems, SIAM J. Math. Anal., 13 (1982), 555-593.
doi: 10.1137/0513037. |
[23] |
J. Sherman and W. J. Morrison, Adjustment of an inverse matrix corresponding to a change in one element of a given matrix, Ann. Math. Statistics, 21 (1950), 124-127. |
[24] |
A. Turing, The chemical basis of morphogenesis, Phil. Trans. R. Soc. Lond. B, 327 (1952), 37-72.
doi: 10.1098/rstb.1952.0012. |
[25] |
J. Wei, On the boundary spike layer solutions to a singularly perturbed neumann problem, J. Diff. Eq., 134 (1997), 104-133.
doi: 10.1006/jdeq.1996.3218. |
[26] |
J. Wei, On single interior spike solutions of the Gierer-Meinhardt system: uniqueness and spectrum estimates, European J. Appl. Math., 10 (1999), 353-378.
doi: 10.1017/S0956792599003770. |
[27] |
J. Wei, On a nonlocal eigenvalue problem and its applications to point-condensations in reaction-diffusion systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 10 (2000), 1485-1496.
doi: 10.1142/S0218127400000979. |
[28] |
J. Wei, Uniqueness and critical spectrum of boundary spike solutions, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 1457-1480.
doi: 10.1017/S0308210500001487. |
[29] |
J. Wei and M. Winter, Spikes for the two-dimensional Gierer-Meinhardt system: the weak coupling case, J. Nonlinear Sci., 11 (2001), 415-458.
doi: 10.1007/s00332-001-0380-1. |
show all references
References:
[1] |
N. D. Alikakos, P. W. Bates, X. Chen and G. Fusco, Mullins-Sekerka motion of small droplets on a fixed boundary, J. Geom. Anal., 10 (2000), 575-596. |
[2] |
P. W. Bates, K. Lu and C. Zeng, Approximately invariant manifolds and global dynamics of spike states, Invent. Math., 174 (2008), 355-433. |
[3] |
X. Chen and M. Kowalczyk, Slow dynamics of interior spikes in the shadow Gierer-Meinhardt system, Adv. Differential Equations, 6 (2001), 847-872. |
[4] |
M. del Pino, P. L. Felmer and M. Kowalczyk, Boundary spikes in the Gierer-Meinhardt system, Commun. Pure Appl. Anal., 1 (2002), 437-456.
doi: 10.3934/cpaa.2002.1.437. |
[5] |
M. del Pino, P. L. Felmer and J. Wei, On the role of mean curvature in some singularly perturbed Neumann problems, SIAM J. Math. Anal., 31 (1999), 63-79 (electronic).
doi: 10.1137/S0036141098332834. |
[6] |
M. del Pino, P. L. Felmer and J. Wei, On the role of distance function in some singular perturbation problems, Comm. Partial Differential Equations, 25 (2000), 155-177.
doi: 10.1080/03605300008821511. |
[7] |
S.-I. Ei, The motion of weakly interacting pulses in reaction-diffusion systems, J. D. D. E. 14 (2002), 85-137.
doi: 10.1023/A:1012980128575. |
[8] |
S.-I. Ei, Dynamics and their interaction of spikes on smoothly curved boundaries for reaction-diffusion systems in 2D, preprint. |
[9] |
L. C. Evans, "Partial Differential Equations,'' Graduate Studies in Mathematics, 19, (1998). |
[10] |
B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$, Mathematical analysis and applications, Part A, Adv. in Math. Suppl. Stud., 12 (1972), 30-39. Academic Press, (1981), 369-402. |
[11] |
A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetic, 12 (1972), 30-39.
doi: 10.1007/BF00289234. |
[12] |
C. Gui and J. Wei, On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems, Canad. J. Math., 52 (2000), 522-538.
doi: 10.4153/CJM-2000-024-x. |
[13] |
D. Iron and M. J. Ward, The dynamics of boundary spikes for a nonlocal reaction-diffusion model, European J. Appl. Math., 11 (2000), 491-514.
doi: 10.1017/S0956792500004253. |
[14] |
D. Iron, M. J. 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. |
[15] |
C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27.
doi: 10.1016/0022-0396(88)90147-7. |
[16] |
H. Meinhardt, "Models of Biological Pattern Formation,'' Academic Press, 1982. |
[17] |
Y. Miyamoto, Stability of a boundary spike layer for the Gierer-Meinhardt system, European J. Appl. Math., 16 (2005), 467-491.
doi: 10.1017/S0956792505006376. |
[18] |
Y. Miyamoto, On the shape of the stable patterns for activator-inhibitor systems in two-dimensional domains, Quarterly of Applied Mathematics, 65 (2007), 357-374. |
[19] |
W.-M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math., 44 (1991), 819-851.
doi: 10.1002/cpa.3160440705. |
[20] |
W.-M. Ni and I. Takagi, Locating the peaks of least energy solution to a semilinear Neumann problem, Duke Math. J., 70 (1993), 247-281.
doi: 10.1215/S0012-7094-93-07004-4. |
[21] |
W.-M. Ni, I. Takagi and E. Yanagida, Stability of least energy patterns of the shadow system for an activator-inhibitor model, Japan J. Indust. Appl. Math., 18, (2001), 259-272. |
[22] |
Y. Nishiura, Global structure of bifurcating solutions of some reaction-diffusion systems, SIAM J. Math. Anal., 13 (1982), 555-593.
doi: 10.1137/0513037. |
[23] |
J. Sherman and W. J. Morrison, Adjustment of an inverse matrix corresponding to a change in one element of a given matrix, Ann. Math. Statistics, 21 (1950), 124-127. |
[24] |
A. Turing, The chemical basis of morphogenesis, Phil. Trans. R. Soc. Lond. B, 327 (1952), 37-72.
doi: 10.1098/rstb.1952.0012. |
[25] |
J. Wei, On the boundary spike layer solutions to a singularly perturbed neumann problem, J. Diff. Eq., 134 (1997), 104-133.
doi: 10.1006/jdeq.1996.3218. |
[26] |
J. Wei, On single interior spike solutions of the Gierer-Meinhardt system: uniqueness and spectrum estimates, European J. Appl. Math., 10 (1999), 353-378.
doi: 10.1017/S0956792599003770. |
[27] |
J. Wei, On a nonlocal eigenvalue problem and its applications to point-condensations in reaction-diffusion systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 10 (2000), 1485-1496.
doi: 10.1142/S0218127400000979. |
[28] |
J. Wei, Uniqueness and critical spectrum of boundary spike solutions, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 1457-1480.
doi: 10.1017/S0308210500001487. |
[29] |
J. Wei and M. Winter, Spikes for the two-dimensional Gierer-Meinhardt system: the weak coupling case, J. Nonlinear Sci., 11 (2001), 415-458.
doi: 10.1007/s00332-001-0380-1. |
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