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Positive solution for the Kirchhoff-type equations involving general subcritical growth
Competing interactions and traveling wave solutions in lattice differential equations
1. | Department of Mathematics, University of Kansas, Lawrence, KS 66045, United States |
2. | Department of Mathematical Sciences, University of Arkansas, Fayetteville, AR 72701, United States |
References:
[1] |
P. W. Bates, X. F. 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] |
M. Brucal - Hallare and E.S. Van Vleck, Traveling fronts in an antidiffusion lattice Nagumo model, SIAM J. Appl. Dyn. Sys., 10 (2011), 921-959.
doi: 10.1137/100819461. |
[3] |
J. W. Cahn, J. Mallet-Paret and E. S. Van Vleck, Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice, SIAM J. Appl. Math., 59 (1999), 455-493.
doi: 10.1137/S0036139996312703. |
[4] |
X. F. Chen, J. S. Guo and C. C. Wu, Traveling waves in discrete periodic media for bistable dynamics, Arch. Ration. Mech. Anal., 189 (2008), 189-236.
doi: 10.1007/s00205-007-0103-3. |
[5] |
S. N. Chow, J. Mallet-Paret and W. X. Shen, Traveling waves in lattice dynamical systems, J. Differential Equations, 149 (1998), 248-291.
doi: 10.1006/jdeq.1998.3478. |
[6] |
J. Harterich, B. Sandstede and A. Scheel, Exponential dichotomies for linear non-autonomous functional differential equations of mixed type, Indiana Univ. Math. J., 51 (2002), 1081-1109.
doi: 10.1512/iumj.2002.51.2188. |
[7] |
H. J. Hupkes and S. M. Verduyn-Lunel, Analysis of Newton's method to compute travelling waves in discrete media, J. Dynam. Differential Equations, 17 (2005), 523-572.
doi: 10.1007/s10884-005-5809-z. |
[8] |
H. J. Hupkes and S. M. Verduyn-Lunel, Center manifold theory for functional differential equations of mixed type, J. Dynam. Diff. Eqns., 19 (2007), 497-560.
doi: 10.1007/s10884-006-9055-9. |
[9] |
H. J. Hupkes and S. M. Verduyn-Lunel, Lin's method and homoclinic bifurcations for functional differential equations of mixed type, Indiana Univ. Math. J., 58 (2009), 2433-2487.
doi: 10.1512/iumj.2009.58.3661. |
[10] |
H. J. Hupkes and B. Sandstede, Traveling pulse solutions for the discrete FitzHugh-Nagumo system, SIAM J. Appl. Dyn. Syst., 9 (2010), 827-882.
doi: 10.1137/090771740. |
[11] |
H. J. Hupkes and E. S. Van Vleck, Negative diffusion and traveling waves in high dimensional lattice systems, SIAM J. Math. Anal., 45 (2013), 1068-1135.
doi: 10.1137/120880628. |
[12] |
C. Lamb and E. S. Van Vleck, Neutral mixed type functional differential equations, J. Dynam. Differential Equations, (2015), in press. |
[13] |
J. Mallet-Paret and S. M. Verduyn-Lunel, Exponential dichotomies and Wiener-Hopf factorizations for mixed-type functional differential equations,, \emph{J. of Differential Equations}, ().
|
[14] |
J. Mallet-Paret, The Fredholm alternative for functional differential equations of mixed type, J. Dynamics and Differential Equations, 11 (1999), 1-48.
doi: 10.1023/A:1021889401235. |
[15] |
J. Mallet-Paret, The global structure of traveling waves in spatially discrete dynamical systems, J. Dyn. Diff. Eqn., 11 (1999), 49-127.
doi: 10.1023/A:1021841618074. |
[16] |
J. Mallet-Paret, Traveling waves in spatially-discrete dynamical systems of diffusive type, Lecture Notes in Math, 1822 (2003), 231-298.
doi: 10.1007/978-3-540-45204-1_4. |
[17] |
A. Rustichini, Functional-differential equations of mixed type: the linear autonomous case, J. Dynam. Differential Equations, 1 (1989), 121-143,
doi: 10.1007/BF01047828. |
[18] |
A. Rustichini, Hopf bifurcation for functional-differential equations of mixed type, J. Dynam. Differential Equations, 1 (1989), 145-177.
doi: 10.1007/BF01047829. |
[19] |
W. Shen, Travelling waves in time almost periodic structures governed by bistable nonlinearities: I. Stability and uniqueness, J. Differential Equations, 159 (1999), 1-54.
doi: 10.1006/jdeq.1999.3651. |
[20] |
W. Shen, Travelling waves in time almost periodic structures governed by bistable nonlinearities: II. Existence, J. Differential Equations, 159 (1999), 55-101.
doi: 10.1006/jdeq.1999.3652. |
[21] |
A. Vainchtein and E. S. Van Vleck, Nucleation and propagation of phase mixtures in a bistable chain, Phys. Rev. B., 79 (2009), 144123. |
[22] |
A. Vainchtein, E. S. Van Vleck and A. Zhang, Propagation of periodic patterns in a discrete system with competing interactions, SIAM J. Appl. Dyn. Sys., 14 (2015), 523-555. |
[23] |
B. Zinner, Stability of traveling wavefronts for the discrete Nagumo equation, SIAM J. Math. Anal., 22 (1991), 1016-1020.
doi: 10.1137/0522066. |
[24] |
B. Zinner, Existence of traveling wavefront solutions for the discrete Nagumo equation, J. Diff. Eqn, 96 (1992), 1-27.
doi: 10.1016/0022-0396(92)90142-A. |
show all references
References:
[1] |
P. W. Bates, X. F. 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] |
M. Brucal - Hallare and E.S. Van Vleck, Traveling fronts in an antidiffusion lattice Nagumo model, SIAM J. Appl. Dyn. Sys., 10 (2011), 921-959.
doi: 10.1137/100819461. |
[3] |
J. W. Cahn, J. Mallet-Paret and E. S. Van Vleck, Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice, SIAM J. Appl. Math., 59 (1999), 455-493.
doi: 10.1137/S0036139996312703. |
[4] |
X. F. Chen, J. S. Guo and C. C. Wu, Traveling waves in discrete periodic media for bistable dynamics, Arch. Ration. Mech. Anal., 189 (2008), 189-236.
doi: 10.1007/s00205-007-0103-3. |
[5] |
S. N. Chow, J. Mallet-Paret and W. X. Shen, Traveling waves in lattice dynamical systems, J. Differential Equations, 149 (1998), 248-291.
doi: 10.1006/jdeq.1998.3478. |
[6] |
J. Harterich, B. Sandstede and A. Scheel, Exponential dichotomies for linear non-autonomous functional differential equations of mixed type, Indiana Univ. Math. J., 51 (2002), 1081-1109.
doi: 10.1512/iumj.2002.51.2188. |
[7] |
H. J. Hupkes and S. M. Verduyn-Lunel, Analysis of Newton's method to compute travelling waves in discrete media, J. Dynam. Differential Equations, 17 (2005), 523-572.
doi: 10.1007/s10884-005-5809-z. |
[8] |
H. J. Hupkes and S. M. Verduyn-Lunel, Center manifold theory for functional differential equations of mixed type, J. Dynam. Diff. Eqns., 19 (2007), 497-560.
doi: 10.1007/s10884-006-9055-9. |
[9] |
H. J. Hupkes and S. M. Verduyn-Lunel, Lin's method and homoclinic bifurcations for functional differential equations of mixed type, Indiana Univ. Math. J., 58 (2009), 2433-2487.
doi: 10.1512/iumj.2009.58.3661. |
[10] |
H. J. Hupkes and B. Sandstede, Traveling pulse solutions for the discrete FitzHugh-Nagumo system, SIAM J. Appl. Dyn. Syst., 9 (2010), 827-882.
doi: 10.1137/090771740. |
[11] |
H. J. Hupkes and E. S. Van Vleck, Negative diffusion and traveling waves in high dimensional lattice systems, SIAM J. Math. Anal., 45 (2013), 1068-1135.
doi: 10.1137/120880628. |
[12] |
C. Lamb and E. S. Van Vleck, Neutral mixed type functional differential equations, J. Dynam. Differential Equations, (2015), in press. |
[13] |
J. Mallet-Paret and S. M. Verduyn-Lunel, Exponential dichotomies and Wiener-Hopf factorizations for mixed-type functional differential equations,, \emph{J. of Differential Equations}, ().
|
[14] |
J. Mallet-Paret, The Fredholm alternative for functional differential equations of mixed type, J. Dynamics and Differential Equations, 11 (1999), 1-48.
doi: 10.1023/A:1021889401235. |
[15] |
J. Mallet-Paret, The global structure of traveling waves in spatially discrete dynamical systems, J. Dyn. Diff. Eqn., 11 (1999), 49-127.
doi: 10.1023/A:1021841618074. |
[16] |
J. Mallet-Paret, Traveling waves in spatially-discrete dynamical systems of diffusive type, Lecture Notes in Math, 1822 (2003), 231-298.
doi: 10.1007/978-3-540-45204-1_4. |
[17] |
A. Rustichini, Functional-differential equations of mixed type: the linear autonomous case, J. Dynam. Differential Equations, 1 (1989), 121-143,
doi: 10.1007/BF01047828. |
[18] |
A. Rustichini, Hopf bifurcation for functional-differential equations of mixed type, J. Dynam. Differential Equations, 1 (1989), 145-177.
doi: 10.1007/BF01047829. |
[19] |
W. Shen, Travelling waves in time almost periodic structures governed by bistable nonlinearities: I. Stability and uniqueness, J. Differential Equations, 159 (1999), 1-54.
doi: 10.1006/jdeq.1999.3651. |
[20] |
W. Shen, Travelling waves in time almost periodic structures governed by bistable nonlinearities: II. Existence, J. Differential Equations, 159 (1999), 55-101.
doi: 10.1006/jdeq.1999.3652. |
[21] |
A. Vainchtein and E. S. Van Vleck, Nucleation and propagation of phase mixtures in a bistable chain, Phys. Rev. B., 79 (2009), 144123. |
[22] |
A. Vainchtein, E. S. Van Vleck and A. Zhang, Propagation of periodic patterns in a discrete system with competing interactions, SIAM J. Appl. Dyn. Sys., 14 (2015), 523-555. |
[23] |
B. Zinner, Stability of traveling wavefronts for the discrete Nagumo equation, SIAM J. Math. Anal., 22 (1991), 1016-1020.
doi: 10.1137/0522066. |
[24] |
B. Zinner, Existence of traveling wavefront solutions for the discrete Nagumo equation, J. Diff. Eqn, 96 (1992), 1-27.
doi: 10.1016/0022-0396(92)90142-A. |
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