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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 and Continuous Dynamical Systems, 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[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.

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