-
Previous Article
A Heteroclinic Solution to a Variational Problem Corresponding to FitzHugh-Nagumo type Reaction-Diffusion System with Heterogeneity
- CPAA Home
- This Issue
-
Next Article
Multiple solutions for a fractional nonlinear Schrödinger equation with local potential
Existence of traveling waves for a class of nonlocal nonlinear equations with bell shaped kernels
1. | Faculty of Engineering and Natural Sciences, Sabanci University, Tuzla 34956, Istanbul, Turkey |
2. | Department of Mathematics, 405 Snow Hall, University of Kansas, Lawrence, KS 66045 |
In this article we are concerned with the existence of traveling wave solutions of a general class of nonlocal wave equations: $~u_{tt}-a^{2}u_{xx}=(β * u^{p})_{xx}$, $~p>1$. Members of the class arise as mathematical models for the propagation of waves in a wide variety of situations. We assume that the kernel $β $ is a bell-shaped function satisfying some mild differentiability and growth conditions. Taking advantage of growth properties of bell-shaped functions, we give a simple proof for the existence of bell-shaped traveling wave solutions.
References:
[1] |
C. Babaoglu, H. A. Erbay and A. Erkip,
Global existence and blow-up of solutions for a general class of doubly dispersive nonlocal nonlinear wave equations, Nonlinear Anal., 77 (2013), 82-93.
doi: 10.1016/j.na.2012.09.001. |
[2] |
N. Duruk, H. A. Erbay and A. Erkip,
Global existence and blow-up for a class of nonlocal nonlinear Cauchy problems arising in elasticity, Nonlinearity, 23 (2010), 107-118.
doi: 10.1088/0951-7715/23/1/006. |
[3] |
J. M. English and R. L. Pego,
On the solitarywave pulse in a chain of beads, Proc. AMS, 133, 23 (2005), 1763-1768.
doi: 10.1090/S0002-9939-05-07851-2. |
[4] |
H. A. Erbay, S. Erbay and A. Erkip,
Existence and stability of traveling waves for a class of nonlocal nonlinear equations, J. Mathematical Analysis and Applications, 425 (2015), 307-336.
doi: 10.1016/j.jmaa.2014.12.039. |
[5] |
G. Friesecke and J. A. D. Wattis,
Existence theorem for solitary waves on lattices, Commun. Math. Phys., 161 (1994), 391-418.
|
[6] |
E. H. Lieb and M. Loss, Analysis Volume 14, Graduate Studies in Mathematics, AMS, 2001.
doi: 10.1090/gsm/014. |
[7] |
P. L. Lions,
The concentration-compactness principle in the calculus of variations. The locally compact case, Part 1, Ann. Inst. Henri Poincaré, 1 (1984), 109-145.
|
[8] |
A. Stefanov and P. Kevrekidis,
On the existence of solitary traveling waves for generalized Hertzian chains, Journal of Nonlinear Science, 22 (2012), 327-349.
doi: 10.1007/s00332-011-9119-9. |
[9] |
A. Stefanov and P. Kevrekidis,
Traveling waves for monomer chains with precompression, Nonlinearity, 26 (2013), 539-349.
doi: 10.1088/0951-7715/26/2/539. |
[10] |
J. Smoller, Shock Waves and Reaction-Diffusion Equations 2nd edition, Springer-Verlag, New York, 1994.
doi: 10.1007/978-1-4612-0873-0. |
[11] |
C. Wolf,
A mathematical model for the propagation of a hantavirus in structured populations, Discrete Continuous Dynam. Systems-B, 4 (2004), 1065-1089.
doi: 10.3934/dcdsb.2004.4.1065. |
show all references
References:
[1] |
C. Babaoglu, H. A. Erbay and A. Erkip,
Global existence and blow-up of solutions for a general class of doubly dispersive nonlocal nonlinear wave equations, Nonlinear Anal., 77 (2013), 82-93.
doi: 10.1016/j.na.2012.09.001. |
[2] |
N. Duruk, H. A. Erbay and A. Erkip,
Global existence and blow-up for a class of nonlocal nonlinear Cauchy problems arising in elasticity, Nonlinearity, 23 (2010), 107-118.
doi: 10.1088/0951-7715/23/1/006. |
[3] |
J. M. English and R. L. Pego,
On the solitarywave pulse in a chain of beads, Proc. AMS, 133, 23 (2005), 1763-1768.
doi: 10.1090/S0002-9939-05-07851-2. |
[4] |
H. A. Erbay, S. Erbay and A. Erkip,
Existence and stability of traveling waves for a class of nonlocal nonlinear equations, J. Mathematical Analysis and Applications, 425 (2015), 307-336.
doi: 10.1016/j.jmaa.2014.12.039. |
[5] |
G. Friesecke and J. A. D. Wattis,
Existence theorem for solitary waves on lattices, Commun. Math. Phys., 161 (1994), 391-418.
|
[6] |
E. H. Lieb and M. Loss, Analysis Volume 14, Graduate Studies in Mathematics, AMS, 2001.
doi: 10.1090/gsm/014. |
[7] |
P. L. Lions,
The concentration-compactness principle in the calculus of variations. The locally compact case, Part 1, Ann. Inst. Henri Poincaré, 1 (1984), 109-145.
|
[8] |
A. Stefanov and P. Kevrekidis,
On the existence of solitary traveling waves for generalized Hertzian chains, Journal of Nonlinear Science, 22 (2012), 327-349.
doi: 10.1007/s00332-011-9119-9. |
[9] |
A. Stefanov and P. Kevrekidis,
Traveling waves for monomer chains with precompression, Nonlinearity, 26 (2013), 539-349.
doi: 10.1088/0951-7715/26/2/539. |
[10] |
J. Smoller, Shock Waves and Reaction-Diffusion Equations 2nd edition, Springer-Verlag, New York, 1994.
doi: 10.1007/978-1-4612-0873-0. |
[11] |
C. Wolf,
A mathematical model for the propagation of a hantavirus in structured populations, Discrete Continuous Dynam. Systems-B, 4 (2004), 1065-1089.
doi: 10.3934/dcdsb.2004.4.1065. |
[1] |
Riccardo Aragona, Marco Calderini, Roberto Civino, Massimiliano Sala, Ilaria Zappatore. Wave-shaped round functions and primitive groups. Advances in Mathematics of Communications, 2019, 13 (1) : 67-88. doi: 10.3934/amc.2019004 |
[2] |
Zalman Balanov, Carlos García-Azpeitia, Wieslaw Krawcewicz. On variational and topological methods in nonlinear difference equations. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2813-2844. doi: 10.3934/cpaa.2018133 |
[3] |
Matthieu Alfaro, Jérôme Coville, Gaël Raoul. Bistable travelling waves for nonlocal reaction diffusion equations. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1775-1791. doi: 10.3934/dcds.2014.34.1775 |
[4] |
Jason Metcalfe, Christopher D. Sogge. Global existence for high dimensional quasilinear wave equations exterior to star-shaped obstacles. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1589-1601. doi: 10.3934/dcds.2010.28.1589 |
[5] |
Mathias Nikolai Arnesen. Existence of solitary-wave solutions to nonlocal equations. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3483-3510. doi: 10.3934/dcds.2016.36.3483 |
[6] |
Lili Fan, Hongxia Liu, Huijiang Zhao, Qingyang Zou. Global stability of stationary waves for damped wave equations. Kinetic and Related Models, 2013, 6 (4) : 729-760. doi: 10.3934/krm.2013.6.729 |
[7] |
Weiran Sun, Min Tang. A relaxation method for one dimensional traveling waves of singular and nonlocal equations. Discrete and Continuous Dynamical Systems - B, 2013, 18 (5) : 1459-1491. doi: 10.3934/dcdsb.2013.18.1459 |
[8] |
Fengxin Chen. Stability and uniqueness of traveling waves for system of nonlocal evolution equations with bistable nonlinearity. Discrete and Continuous Dynamical Systems, 2009, 24 (3) : 659-673. doi: 10.3934/dcds.2009.24.659 |
[9] |
Peter Giesl, Sigurdur Hafstein. Computational methods for Lyapunov functions. Discrete and Continuous Dynamical Systems - B, 2015, 20 (8) : i-ii. doi: 10.3934/dcdsb.2015.20.8i |
[10] |
Luc Molinet, Francis Ribaud. On global well-posedness for a class of nonlocal dispersive wave equations. Discrete and Continuous Dynamical Systems, 2006, 15 (2) : 657-668. doi: 10.3934/dcds.2006.15.657 |
[11] |
H. A. Erbay, S. Erbay, A. Erkip. Long-time existence of solutions to nonlocal nonlinear bidirectional wave equations. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2877-2891. doi: 10.3934/dcds.2019119 |
[12] |
Zaihui Gan. Cross-constrained variational methods for the nonlinear Klein-Gordon equations with an inverse square potential. Communications on Pure and Applied Analysis, 2009, 8 (5) : 1541-1554. doi: 10.3934/cpaa.2009.8.1541 |
[13] |
F. Ali Mehmeti, R. Haller-Dintelmann, V. Régnier. Dispersive waves with multiple tunnel effect on a star-shaped network. Discrete and Continuous Dynamical Systems - S, 2013, 6 (3) : 783-791. doi: 10.3934/dcdss.2013.6.783 |
[14] |
H. A. Erbay, S. Erbay, A. Erkip. The Camassa-Holm equation as the long-wave limit of the improved Boussinesq equation and of a class of nonlocal wave equations. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6101-6116. doi: 10.3934/dcds.2016066 |
[15] |
Lijin Wang, Jialin Hong. Generating functions for stochastic symplectic methods. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 1211-1228. doi: 10.3934/dcds.2014.34.1211 |
[16] |
Peter Giesl, Sigurdur Hafstein. Review on computational methods for Lyapunov functions. Discrete and Continuous Dynamical Systems - B, 2015, 20 (8) : 2291-2331. doi: 10.3934/dcdsb.2015.20.2291 |
[17] |
Farah Abou Shakra. Asymptotics of wave models for non star-shaped geometries. Discrete and Continuous Dynamical Systems - S, 2014, 7 (2) : 347-362. doi: 10.3934/dcdss.2014.7.347 |
[18] |
Ahmed Bchatnia, Amina Boukhatem. Stability of a damped wave equation on an infinite star-shaped network. Evolution Equations and Control Theory, 2022 doi: 10.3934/eect.2022024 |
[19] |
Rui Huang, Ming Mei, Kaijun Zhang, Qifeng Zhang. Asymptotic stability of non-monotone traveling waves for time-delayed nonlocal dispersion equations. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1331-1353. doi: 10.3934/dcds.2016.36.1331 |
[20] |
Yicheng Jiang, Kaijun Zhang. Stability of traveling waves for nonlocal time-delayed reaction-diffusion equations. Kinetic and Related Models, 2018, 11 (5) : 1235-1253. doi: 10.3934/krm.2018048 |
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]