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Optimal regularity and stability analysis in the $\alpha-$Norm for a class of partial functional differential equations with infinite delay
Existence and uniqueness of traveling waves in a class of unidirectional lattice differential equations
1. | Franklin W. Olin College of Engineering, 1000 Olin Way, Needham, MA 02492-1200, United States |
2. | Gettysburg College, Department of Mathematics, 300 North Washington St., Gettysburg, PA 17325-1400, United States |
References:
[1] |
P. Bates, X. Chen and A. Chmaj, Traveling waves of bistable dynamics on a lattice, SIAM J. Math. Anal., 35 (2003), 520-546.
doi: 10.1137/S0036141000374002. |
[2] |
S. Benzoni-Gavage, Semi-discrete shock profiles for hyperbolic systems of conservation laws, Phys. D, 115 (1998), 109-123.
doi: 10.1016/S0167-2789(97)00225-X. |
[3] |
S. Bianchini, BV solutions of semidiscrete upwind scheme, in "Hyperbolic problems: Theory, Numerics, Applications," Springer, (2003), 135-142. |
[4] |
J. W. Cahn, J. Mallet-Paret and E. Van Vleck, Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice, SIAM J. Appl. Math, 59 (1999), 455-493. |
[5] |
J. O. Chua and L. Yang, Cellular neural networks: Theory, IEEE Trans. Circuits and Systems, 35 (1998), 1257-1272.
doi: 10.1109/31.7600. |
[6] |
J. O. Chua and L. Yang, Cellular neural networks: applications, IEEE Trans. Circuits and Systems, 35 (1998), 1273-1290.
doi: 10.1109/31.7601. |
[7] |
J. L. Daleckii and M. G. Krein, "Stability of Solutions of Differential Equations in Banach Space,", Translated from the Russian by S.Smith, 43 ().
|
[8] |
O. Diekmann, S. A. van Gils, S. M. Verduyn-Lunel and H.-O. Walther, "Delay Equations: Functional- Complex- and Nonlinear Analysis," volume 110 of Applied Mathematical Sciences, Springer-Verlag, New York 1995. |
[9] |
U. Ebert, W. Van Saarloos and L. A. Peletier, Universal algebraic convergence in time of pulled fronts: the common mechanism for difference-differential and partial differential equations, European J. Appl. Math, 13 (2002), 53-66.
doi: 10.1017/S0956792501004673. |
[10] |
R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 335-369. |
[11] |
J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional-Differential Equations," volume 99 of Applied Mathematical Sciences, Springer-Verlag, New York, 1993. |
[12] |
D. Henry, Small solutions of linear autonomous functional differential equations, J. Differential Equations, 8 (1970), 494-501.
doi: 10.1016/0022-0396(70)90021-5. |
[13] |
A. Hoffman and J. Mallet-Paret, Universality of crystallographic pinning, J. Dynam. Differential Equations, 22 (2010), 79-119.
doi: 10.1007/s10884-010-9157-2. |
[14] |
S.-S. Hsu and C.-H. Lin, Existence and multiplicity of traveling waves in a lattice dynamical system, J. Differential Equations, 164 (2000), 431-450.
doi: 10.1006/jdeq.2000.3770. |
[15] |
S.-S. Hsu, C.-H. Lin and W. Shen, Traveling waves in cellular neural networks, International Journal of Bifurcation and Chaos, 9 (1999), 1307-1319.
doi: 10.1142/S0218127499000912. |
[16] |
S.-B. Hsu and X.-Q. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations, SIAM J. Math. Anal., 40 (2008), 776-789.
doi: 10.1137/070703016. |
[17] |
J. Keener, Propagation and its failure in coupled systems of discrete excitable cells, SIAM J. Appl. Math., 47 (1987), 556-572.
doi: 10.1137/0147038. |
[18] |
A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Etude de l'equation de la diffusion avec croissance de la quantite de matiere et som application a un probleme biologique, Bull. Universite d'Etat a Moscou Ser. Int., Sect. A., 1 (1937), 1-25. |
[19] |
T. Krisztin, Global dynamics of delay differential equations, Period. Math. Hungar., 56 (2008), 83-95.
doi: 10.1007/s10998-008-5083-x. |
[20] |
B. Li, M. A. Lewis and H. F. Weinberger, Existence of traveling waves for integral recursions with nonmonotone growth functions, J. Math. Biol., 58 (2009), 323-338.
doi: 10.1007/s00285-008-0175-1. |
[21] |
J. Mallet-Paret, Morse decompositions for delay-differential equations, J. Differential Equations, 72 (1988), 270-315.
doi: 10.1016/0022-0396(88)90157-X. |
[22] |
J. Mallet-Paret, The Fredholm alternative for functional-differential equations of mixed type, J. Dynam. Differential Equations, 11 (1999), 1-47.
doi: 10.1023/A:1021889401235. |
[23] |
J. Mallet-Paret, Traveling waves in spatially discrete dynamical systems of diffusive type, in "Dynamical Systems," volume 1822 of Lecture Notes in Math., Springer, (2003), 231-298. |
[24] |
J. Mallet-Paret and G. Sell, The Poincare-Bendixson theorem for monotone cyclic feedback systems with delay, J. Differential Equations, 125 (1996), 441-489.
doi: 10.1006/jdeq.1996.0037. |
[25] |
C. Mascia, Qualitative behavior of conservation laws with reaction term and nonconvex flux, Quart. Appl. Math., 58 (2000), 739-761. |
[26] |
R. D. Nussbaum, Periodic solutions of some nonlinear autonomous functional differential equations, Ann. Mat. Pura Appl., 4 (1974), 263-306. |
[27] |
L. A. Peletier and J. A. Rodriguez, Fronts on a lattice, Differential Integral Equations, 17 (2004), 1013-1042. |
[28] |
W. Rudin, "Principles of Mathematical Analysis," McGraw-Hill, 1976. |
[29] |
D. Serre, Discrete shock profiles: Existence and stability, in "Hyperbolic Systems of Balance Laws," volume 1911 of Lecture Notes in Math., Springer, (2007), 79-158. |
[30] |
W. van Saarloos, Front Propagation into unstable states, Phys. Rep., 386 (2003), 29-222.
doi: 10.1016/j.physrep.2003.08.001. |
[31] |
R. van Zon, H. van Beijeren and Ch. Dellago, Largest Lyapunov exponent for many particle systems at low densities, Phys. Rev. Lett., 80 (1998), 2035-2038.
doi: 10.1103/PhysRevLett.80.2035. |
[32] |
H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396.
doi: 10.1137/0513028. |
show all references
References:
[1] |
P. Bates, X. Chen and A. Chmaj, Traveling waves of bistable dynamics on a lattice, SIAM J. Math. Anal., 35 (2003), 520-546.
doi: 10.1137/S0036141000374002. |
[2] |
S. Benzoni-Gavage, Semi-discrete shock profiles for hyperbolic systems of conservation laws, Phys. D, 115 (1998), 109-123.
doi: 10.1016/S0167-2789(97)00225-X. |
[3] |
S. Bianchini, BV solutions of semidiscrete upwind scheme, in "Hyperbolic problems: Theory, Numerics, Applications," Springer, (2003), 135-142. |
[4] |
J. W. Cahn, J. Mallet-Paret and E. Van Vleck, Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice, SIAM J. Appl. Math, 59 (1999), 455-493. |
[5] |
J. O. Chua and L. Yang, Cellular neural networks: Theory, IEEE Trans. Circuits and Systems, 35 (1998), 1257-1272.
doi: 10.1109/31.7600. |
[6] |
J. O. Chua and L. Yang, Cellular neural networks: applications, IEEE Trans. Circuits and Systems, 35 (1998), 1273-1290.
doi: 10.1109/31.7601. |
[7] |
J. L. Daleckii and M. G. Krein, "Stability of Solutions of Differential Equations in Banach Space,", Translated from the Russian by S.Smith, 43 ().
|
[8] |
O. Diekmann, S. A. van Gils, S. M. Verduyn-Lunel and H.-O. Walther, "Delay Equations: Functional- Complex- and Nonlinear Analysis," volume 110 of Applied Mathematical Sciences, Springer-Verlag, New York 1995. |
[9] |
U. Ebert, W. Van Saarloos and L. A. Peletier, Universal algebraic convergence in time of pulled fronts: the common mechanism for difference-differential and partial differential equations, European J. Appl. Math, 13 (2002), 53-66.
doi: 10.1017/S0956792501004673. |
[10] |
R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 335-369. |
[11] |
J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional-Differential Equations," volume 99 of Applied Mathematical Sciences, Springer-Verlag, New York, 1993. |
[12] |
D. Henry, Small solutions of linear autonomous functional differential equations, J. Differential Equations, 8 (1970), 494-501.
doi: 10.1016/0022-0396(70)90021-5. |
[13] |
A. Hoffman and J. Mallet-Paret, Universality of crystallographic pinning, J. Dynam. Differential Equations, 22 (2010), 79-119.
doi: 10.1007/s10884-010-9157-2. |
[14] |
S.-S. Hsu and C.-H. Lin, Existence and multiplicity of traveling waves in a lattice dynamical system, J. Differential Equations, 164 (2000), 431-450.
doi: 10.1006/jdeq.2000.3770. |
[15] |
S.-S. Hsu, C.-H. Lin and W. Shen, Traveling waves in cellular neural networks, International Journal of Bifurcation and Chaos, 9 (1999), 1307-1319.
doi: 10.1142/S0218127499000912. |
[16] |
S.-B. Hsu and X.-Q. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations, SIAM J. Math. Anal., 40 (2008), 776-789.
doi: 10.1137/070703016. |
[17] |
J. Keener, Propagation and its failure in coupled systems of discrete excitable cells, SIAM J. Appl. Math., 47 (1987), 556-572.
doi: 10.1137/0147038. |
[18] |
A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Etude de l'equation de la diffusion avec croissance de la quantite de matiere et som application a un probleme biologique, Bull. Universite d'Etat a Moscou Ser. Int., Sect. A., 1 (1937), 1-25. |
[19] |
T. Krisztin, Global dynamics of delay differential equations, Period. Math. Hungar., 56 (2008), 83-95.
doi: 10.1007/s10998-008-5083-x. |
[20] |
B. Li, M. A. Lewis and H. F. Weinberger, Existence of traveling waves for integral recursions with nonmonotone growth functions, J. Math. Biol., 58 (2009), 323-338.
doi: 10.1007/s00285-008-0175-1. |
[21] |
J. Mallet-Paret, Morse decompositions for delay-differential equations, J. Differential Equations, 72 (1988), 270-315.
doi: 10.1016/0022-0396(88)90157-X. |
[22] |
J. Mallet-Paret, The Fredholm alternative for functional-differential equations of mixed type, J. Dynam. Differential Equations, 11 (1999), 1-47.
doi: 10.1023/A:1021889401235. |
[23] |
J. Mallet-Paret, Traveling waves in spatially discrete dynamical systems of diffusive type, in "Dynamical Systems," volume 1822 of Lecture Notes in Math., Springer, (2003), 231-298. |
[24] |
J. Mallet-Paret and G. Sell, The Poincare-Bendixson theorem for monotone cyclic feedback systems with delay, J. Differential Equations, 125 (1996), 441-489.
doi: 10.1006/jdeq.1996.0037. |
[25] |
C. Mascia, Qualitative behavior of conservation laws with reaction term and nonconvex flux, Quart. Appl. Math., 58 (2000), 739-761. |
[26] |
R. D. Nussbaum, Periodic solutions of some nonlinear autonomous functional differential equations, Ann. Mat. Pura Appl., 4 (1974), 263-306. |
[27] |
L. A. Peletier and J. A. Rodriguez, Fronts on a lattice, Differential Integral Equations, 17 (2004), 1013-1042. |
[28] |
W. Rudin, "Principles of Mathematical Analysis," McGraw-Hill, 1976. |
[29] |
D. Serre, Discrete shock profiles: Existence and stability, in "Hyperbolic Systems of Balance Laws," volume 1911 of Lecture Notes in Math., Springer, (2007), 79-158. |
[30] |
W. van Saarloos, Front Propagation into unstable states, Phys. Rep., 386 (2003), 29-222.
doi: 10.1016/j.physrep.2003.08.001. |
[31] |
R. van Zon, H. van Beijeren and Ch. Dellago, Largest Lyapunov exponent for many particle systems at low densities, Phys. Rev. Lett., 80 (1998), 2035-2038.
doi: 10.1103/PhysRevLett.80.2035. |
[32] |
H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396.
doi: 10.1137/0513028. |
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