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November 2017, 16(6): 2125-2132. doi: 10.3934/cpaa.2017105

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

* Corresponding author

Received  December 2016 Revised  February 2017 Published  July 2017

Fund Project: The second author is supported by TUBITAK Grant 2215 -Graduate Scholarship Programme for International Students

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.

Citation: Albert Erkip, Abba I. Ramadan. Existence of traveling waves for a class of nonlocal nonlinear equations with bell shaped kernels. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2125-2132. doi: 10.3934/cpaa.2017105
References:
[1]

C. BabaogluH. 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. DurukH. 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. ErbayS. 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. BabaogluH. 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. DurukH. 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. ErbayS. 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.

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