# American Institute of Mathematical Sciences

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

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

 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.

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] 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 [19] 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 [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