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Explicit upper and lower bounds for the traveling wave solutions of Fisher-Kolmogorov type equations
1. | Dept. de Matemàtiques, Universitat Autònoma de Barcelona, Edifici C, 08193 Bellaterra, Barcelona |
2. | Laboratoire de Mathématique et Physique Théorique, C.N.R.S. UMR 7350, Faculté des Sciences et Techniques, Université de Tours, Parc de Grandmont,37200 Tours |
3. | Departament de Matemàtiques, Universitat Autònoma de Barcelona, Edifici C, 08193 Bellaterra, Barcelona, Spain |
References:
[1] |
M. J. Ablowitz and A. Zeppetella, Explicit solutions of Fisher's equation for a special wave speed, Bull. Math. Biol., 41 (1979), 835-840.
doi: 10.1016/S0092-8240(79)80020-8. |
[2] |
D. G. Aronson and H. F. Weinberger, "Nonlinear Diffusion in Population Genetics, Combustion, and Nerve Pulse Propagation," Partial Differential Equations and Related Topics (Program, Tulane Univ., New Orleans, La., 1974), 5-49. Lecture Notes in Math., 446, Springer, Berlin, 1975. |
[3] |
R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 355-369. |
[4] |
A. Gasull, H. Giacomini and J. Torregrosa, Some results on homoclinic and heteroclinic connections in planar systems, Nonlinearity, 23 (2010), 2977-3001.
doi: 10.1088/0951-7715/23/12/001. |
[5] |
A. Goriely, A simple solution to the nonlinear front problem, Phys. Rev. Lett., 75 (1995), 2047-2050. |
[6] |
P. Grindrod, Patterns and Waves, "The Theory and Applications of Reaction-Diffusion Equations," Clarendon Press, 1991. |
[7] |
A. Kolmogorov, I. Petrovskii and N. Piskunov, A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem, in "Selected Works of A. N. Kolmogorov I" (editor, V. M. Tikhomirov), 248-270. Kluwer 1991. Translated by V. M. Volosov from Bull. Moscow Univ., Math. Mech., 1 (1937), 1-25. |
[8] |
A. C. Newell and J. A. Whitehead, Finite bandwidth, finite amplitude convection, J. Fluid Mech., 38 (1969), 279-303. |
[9] |
M. R. Rodrigo and R. M. Miura, Exact and approximate traveling waves of reaction-diffusion systems via a variational approach, Anal. Appl. (Singap.), 9 (2011), 187-199.
doi: 10.1142/S0219530511001807. |
[10] |
F. Sánchez-Garduño and P. K. Maini, Travelling wave phenomena in some degenerate reaction-diffusion equations, J. Differential Equations, 117 (1995), 281-319.
doi: 10.1006/jdeq.1995.1055. |
[11] |
F. Sánchez-Garduño and P. K. Maini, Travelling wave phenomena in non-linear diffusion degenerate Nagumo equations, J. Math. Biol., 35 (1997), 713-728.
doi: 10.1007/s002850050073. |
[12] |
L. A. Segel, Distant sidewalls cause slow amplitude modulation of cellular convection, J. Fluid Mech., 38 (1969), 203-224. |
[13] |
J. Xin, Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161-230.
doi: 10.1137/S0036144599364296. |
[14] |
Y. B. Zeldovich and D. A. Frank-Kamenetskii, A theory of thermal propagation of flame, Acta Physicochimica URSS 9 (1938), 341-350. English Translation: Dynamics of Curved Fronts, editor P. Pelcé, Perspectives in Physics, Academic Press, New York, (1988), 131-140. |
show all references
References:
[1] |
M. J. Ablowitz and A. Zeppetella, Explicit solutions of Fisher's equation for a special wave speed, Bull. Math. Biol., 41 (1979), 835-840.
doi: 10.1016/S0092-8240(79)80020-8. |
[2] |
D. G. Aronson and H. F. Weinberger, "Nonlinear Diffusion in Population Genetics, Combustion, and Nerve Pulse Propagation," Partial Differential Equations and Related Topics (Program, Tulane Univ., New Orleans, La., 1974), 5-49. Lecture Notes in Math., 446, Springer, Berlin, 1975. |
[3] |
R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 355-369. |
[4] |
A. Gasull, H. Giacomini and J. Torregrosa, Some results on homoclinic and heteroclinic connections in planar systems, Nonlinearity, 23 (2010), 2977-3001.
doi: 10.1088/0951-7715/23/12/001. |
[5] |
A. Goriely, A simple solution to the nonlinear front problem, Phys. Rev. Lett., 75 (1995), 2047-2050. |
[6] |
P. Grindrod, Patterns and Waves, "The Theory and Applications of Reaction-Diffusion Equations," Clarendon Press, 1991. |
[7] |
A. Kolmogorov, I. Petrovskii and N. Piskunov, A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem, in "Selected Works of A. N. Kolmogorov I" (editor, V. M. Tikhomirov), 248-270. Kluwer 1991. Translated by V. M. Volosov from Bull. Moscow Univ., Math. Mech., 1 (1937), 1-25. |
[8] |
A. C. Newell and J. A. Whitehead, Finite bandwidth, finite amplitude convection, J. Fluid Mech., 38 (1969), 279-303. |
[9] |
M. R. Rodrigo and R. M. Miura, Exact and approximate traveling waves of reaction-diffusion systems via a variational approach, Anal. Appl. (Singap.), 9 (2011), 187-199.
doi: 10.1142/S0219530511001807. |
[10] |
F. Sánchez-Garduño and P. K. Maini, Travelling wave phenomena in some degenerate reaction-diffusion equations, J. Differential Equations, 117 (1995), 281-319.
doi: 10.1006/jdeq.1995.1055. |
[11] |
F. Sánchez-Garduño and P. K. Maini, Travelling wave phenomena in non-linear diffusion degenerate Nagumo equations, J. Math. Biol., 35 (1997), 713-728.
doi: 10.1007/s002850050073. |
[12] |
L. A. Segel, Distant sidewalls cause slow amplitude modulation of cellular convection, J. Fluid Mech., 38 (1969), 203-224. |
[13] |
J. Xin, Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161-230.
doi: 10.1137/S0036144599364296. |
[14] |
Y. B. Zeldovich and D. A. Frank-Kamenetskii, A theory of thermal propagation of flame, Acta Physicochimica URSS 9 (1938), 341-350. English Translation: Dynamics of Curved Fronts, editor P. Pelcé, Perspectives in Physics, Academic Press, New York, (1988), 131-140. |
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