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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 |
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.
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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