American Institute of Mathematical Sciences

Advanced
April  2017, 37(4): 1857-1865. doi: 10.3934/dcds.2017078

Bifurcation at infinity for a semilinear wave equation with non-monotone nonlinearity

José Caicedo 1, , Alfonso Castro 2, and  Arturo Sanjuán 3,

1. 

Departamento de Matemáticas, Universidad Nacional de Colombia, Bogotá, Colombia
2. 

Department of Mathematics, Harvey Mudd College, Claremont, CA 91711, USA
3. 

Proyecto Curricular de Matemáticas, Universidad Distrital Francisco José de Caldas, Bogotá, Colombia

Received  May 2016 Revised  November 2016 Published  December 2016

Fund Project: This work was partially supported by a grant from the Simons Foundations (# 245966 to Alfonso Castro)

We prove bifurcation at infinity for a semilinear wave equation depending on a parameter $λ$ and subject to Dirichlet-periodic boundary conditions. We assume the nonlinear term to be asymptotically linear and not necessarily monotone. We prove the existence of L∞ solutions tending to $+∞$ when the bifurcation parameter approaches eigenvalues of finite multiplicity of the wave operator. Further details are presented in cases of simple eigenvalues and odd multiplicity eigenvalues.

Keywords: Semilinear wave equation, non-monotone nonlinearity, bifurcation at infinity, Dirichlet-periodic conditions.
Mathematics Subject Classification: Primary:35L75;Secondary:34B16.
Citation: José Caicedo, Alfonso Castro, Arturo Sanjuán. Bifurcation at infinity for a semilinear wave equation with non-monotone nonlinearity. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1857-1865. doi: 10.3934/dcds.2017078
References:
[1]

H. Brézis, JM. Coron and L. Nirenberg, Free Vibrations for a Nonlinear Wave Equation and a Theorem of P. Rabinowitz, Communications on Pure and Applied Mathematics, 33 (1980), 667-684.  doi: 10.1002/cpa.3160330507.  Google Scholar

[2]

J. F. Caicedo and A. Castro, A Semilinear Wave Equation with Derivative of Nonlinearity Containing Multiple Eigenvalues of Infinite Multiplicity, Contemporary Mathematics, (1997).  doi: 10.1090/conm/208/02737.  Google Scholar

[3]

JF. Caicedo and A. Castro, A semilinear wave equation with smooth data and no resonance having no continuous solution, Continuous and Discrete Dynamical Systems, 24 (2009), 653-658.  doi: 10.3934/dcds.2009.24.653.  Google Scholar

[4]

JF. Caicedo, A. Castro and R. Duque, Existence of Solutions for a wave equation with non-monotone nonlinearity, Milan J. Math, 79 (2011), 207-220.  doi: 10.1007/s00032-011-0154-7.  Google Scholar

[5]

JF. Caicedo, A. Castro, R. Duque and A. Sanjuán, Existence of $L^p$-solutions for a semilinear wave equation with non-monotone nonlinearity, Discrete and Continuous Dynamical Systems Serie S, 7 (2014), 1193-1202.  doi: 10.3934/dcdss.2014.7.1193.  Google Scholar

[6]

A. Castro and B. Preskill, Existence of solutions for a wave equation with non-monotone nonlinearity, Discrete and Continuous Dynamical Systems, 28 (2010), 649-658.  doi: 10.3934/dcds.2010.28.649.  Google Scholar

[7]

A. Castro and S. Unsurangsie, A Semilinear Wave Equation with Nonmonotone Nonlinearity, Pacific Journal of Mathematics, 132 (1988), 215-225.  doi: 10.2140/pjm.1988.132.215.  Google Scholar

[8]

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, 1985, URL http://books.google.com.co/books?id=RYo_AQAAIAAJ. doi: 10.1007/978-3-662-00547-7.  Google Scholar

[9]

N. Fontes-Merz, A multidimensional version of Turán's lemma, Journal of Approximation Theory, 140 (2006), 27-30, URL http://www.sciencedirect.com/science/article/pii/S0021904505002340. doi: 10.1016/j.jat.2005.11.012.  Google Scholar

[10]

H. Hofer, On the range of a wave operator with nonmonotone nonlinearity, Math. Nachr., 106 (1982), 327-340.  doi: 10.1002/mana.19821060128.  Google Scholar

[11]

P. Rabinowitz, Free Vibrations for a Semilinear Wave Equation, Communications on Pure and Applied Mathematics, 31 (1978), 31-68.  doi: 10.1002/cpa.3160310103.  Google Scholar

[12]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, Journal of Functional Analysis, 7 (1971), 487-513, URL http://www.sciencedirect.com/science/article/pii/0022123671900309 doi: 10.1016/0022-1236(71)90030-9.  Google Scholar

[13]

S. Vinogradov, V. Khavin and N. Nikol'skij, Complex Analysis, Operators, and Related Topics: The S. A. Vinogradov Memorial Volume, Operator theory, Springer, 2000, URL http://books.google.com.co/books?id=0SimOIvN3NsC. doi: 10.1007/978-3-0348-8378-8.  Google Scholar

[14]

M. Willem, Density of the range of potential operators, Proceedings of the American Mathematical Society, 83 (1981), 341-344.  doi: 10.1090/S0002-9939-1981-0624926-7.  Google Scholar

show all references

References:
[1]

H. Brézis, JM. Coron and L. Nirenberg, Free Vibrations for a Nonlinear Wave Equation and a Theorem of P. Rabinowitz, Communications on Pure and Applied Mathematics, 33 (1980), 667-684.  doi: 10.1002/cpa.3160330507.  Google Scholar

[2]

J. F. Caicedo and A. Castro, A Semilinear Wave Equation with Derivative of Nonlinearity Containing Multiple Eigenvalues of Infinite Multiplicity, Contemporary Mathematics, (1997).  doi: 10.1090/conm/208/02737.  Google Scholar

[3]

JF. Caicedo and A. Castro, A semilinear wave equation with smooth data and no resonance having no continuous solution, Continuous and Discrete Dynamical Systems, 24 (2009), 653-658.  doi: 10.3934/dcds.2009.24.653.  Google Scholar

[4]

JF. Caicedo, A. Castro and R. Duque, Existence of Solutions for a wave equation with non-monotone nonlinearity, Milan J. Math, 79 (2011), 207-220.  doi: 10.1007/s00032-011-0154-7.  Google Scholar

[5]

JF. Caicedo, A. Castro, R. Duque and A. Sanjuán, Existence of $L^p$-solutions for a semilinear wave equation with non-monotone nonlinearity, Discrete and Continuous Dynamical Systems Serie S, 7 (2014), 1193-1202.  doi: 10.3934/dcdss.2014.7.1193.  Google Scholar

[6]

A. Castro and B. Preskill, Existence of solutions for a wave equation with non-monotone nonlinearity, Discrete and Continuous Dynamical Systems, 28 (2010), 649-658.  doi: 10.3934/dcds.2010.28.649.  Google Scholar

[7]

A. Castro and S. Unsurangsie, A Semilinear Wave Equation with Nonmonotone Nonlinearity, Pacific Journal of Mathematics, 132 (1988), 215-225.  doi: 10.2140/pjm.1988.132.215.  Google Scholar

[8]

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, 1985, URL http://books.google.com.co/books?id=RYo_AQAAIAAJ. doi: 10.1007/978-3-662-00547-7.  Google Scholar

[9]

N. Fontes-Merz, A multidimensional version of Turán's lemma, Journal of Approximation Theory, 140 (2006), 27-30, URL http://www.sciencedirect.com/science/article/pii/S0021904505002340. doi: 10.1016/j.jat.2005.11.012.  Google Scholar

[10]

H. Hofer, On the range of a wave operator with nonmonotone nonlinearity, Math. Nachr., 106 (1982), 327-340.  doi: 10.1002/mana.19821060128.  Google Scholar

[11]

P. Rabinowitz, Free Vibrations for a Semilinear Wave Equation, Communications on Pure and Applied Mathematics, 31 (1978), 31-68.  doi: 10.1002/cpa.3160310103.  Google Scholar

[12]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, Journal of Functional Analysis, 7 (1971), 487-513, URL http://www.sciencedirect.com/science/article/pii/0022123671900309 doi: 10.1016/0022-1236(71)90030-9.  Google Scholar

[13]

S. Vinogradov, V. Khavin and N. Nikol'skij, Complex Analysis, Operators, and Related Topics: The S. A. Vinogradov Memorial Volume, Operator theory, Springer, 2000, URL http://books.google.com.co/books?id=0SimOIvN3NsC. doi: 10.1007/978-3-0348-8378-8.  Google Scholar

[14]

M. Willem, Density of the range of potential operators, Proceedings of the American Mathematical Society, 83 (1981), 341-344.  doi: 10.1090/S0002-9939-1981-0624926-7.  Google Scholar

[1]

Alfonso Castro, Benjamin Preskill. Existence of solutions for a semilinear wave equation with non-monotone nonlinearity. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 649-658. doi: 10.3934/dcds.2010.28.649
[2]

José Caicedo, Alfonso Castro, Rodrigo Duque, Arturo Sanjuán. Existence of $L^p$-solutions for a semilinear wave equation with non-monotone nonlinearity. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1193-1202. doi: 10.3934/dcdss.2014.7.1193
[3]

Pablo Amster, Manuel Zamora. Periodic solutions for indefinite singular equations with singularities in the spatial variable and non-monotone nonlinearity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 4819-4835. doi: 10.3934/dcds.2018211
[4]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825
[5]

Sergiu Aizicovici, Simeon Reich. Anti-periodic solutions to a class of non-monotone evolution equations. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 35-42. doi: 10.3934/dcds.1999.5.35
[6]

Anatoli F. Ivanov, Bernhard Lani-Wayda. Periodic solutions for three-dimensional non-monotone cyclic systems with time delays. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 667-692. doi: 10.3934/dcds.2004.11.667
[7]

Jun Chen, Wenyu Sun, Zhenghao Yang. A non-monotone retrospective trust-region method for unconstrained optimization. Journal of Industrial & Management Optimization, 2013, 9 (4) : 919-944. doi: 10.3934/jimo.2013.9.919
[8]

Andrzej Nowakowski. Variational approach to stability of semilinear wave equation with nonlinear boundary conditions. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2603-2616. doi: 10.3934/dcdsb.2014.19.2603
[9]

Satoshi Kosugi, Yoshihisa Morita, Shoji Yotsutani. A complete bifurcation diagram of the Ginzburg-Landau equation with periodic boundary conditions. Communications on Pure & Applied Analysis, 2005, 4 (3) : 665-682. doi: 10.3934/cpaa.2005.4.665
[10]

Zhao-Xing Yang, Guo-Bao Zhang, Ge Tian, Zhaosheng Feng. Stability of non-monotone non-critical traveling waves in discrete reaction-diffusion equations with time delay. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 581-603. doi: 10.3934/dcdss.2017029
[11]

Abraham Solar. Stability of non-monotone and backward waves for delay non-local reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5799-5823. doi: 10.3934/dcds.2019255
[12]

Rui Huang, Ming Mei, Kaijun Zhang, Qifeng Zhang. Asymptotic stability of non-monotone traveling waves for time-delayed nonlocal dispersion equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1331-1353. doi: 10.3934/dcds.2016.36.1331
[13]

Q-Heung Choi, Tacksun Jung. A nonlinear wave equation with jumping nonlinearity. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 797-802. doi: 10.3934/dcds.2000.6.797
[14]

Patrizia Donato, Florian Gaveau. Homogenization and correctors for the wave equation in non periodic perforated domains. Networks & Heterogeneous Media, 2008, 3 (1) : 97-124. doi: 10.3934/nhm.2008.3.97
[15]

Nikos I. Karachalios, Athanasios N Lyberopoulos. On the dynamics of a degenerate damped semilinear wave equation in \mathbb{R}^N : the non-compact case. Conference Publications, 2007, 2007 (Special) : 531-540. doi: 10.3934/proc.2007.2007.531
[16]

Asma Azaiez. Refined regularity for the blow-up set at non characteristic points for the vector-valued semilinear wave equation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2397-2408. doi: 10.3934/cpaa.2019108
[17]

Micol Amar. A note on boundary layer effects in periodic homogenization with Dirichlet boundary conditions. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 537-556. doi: 10.3934/dcds.2000.6.537
[18]

Zongming Guo, Yunting Yu. Boundary value problems for a semilinear elliptic equation with singular nonlinearity. Communications on Pure & Applied Analysis, 2016, 15 (2) : 399-412. doi: 10.3934/cpaa.2016.15.399
[19]

Yoshiki Maeda, Noboru Okazawa. Schrödinger type evolution equations with monotone nonlinearity of non-power type. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 771-781. doi: 10.3934/dcdss.2013.6.771
[20]

Victor S. Kozyakin, Alexander M. Krasnosel’skii, Dmitrii I. Rachinskii. Arnold tongues for bifurcation from infinity. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 107-116. doi: 10.3934/dcdss.2008.1.107

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (13)
  • HTML views (4)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences