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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 and Applied Analysis, 2017, 16 (6) : 2125-2132. doi: 10.3934/cpaa.2017105
##### 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.
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