Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

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

Pages: 1857 - 1865, Volume 37, Issue 4, April 2017      doi:10.3934/dcds.2017078

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José Caicedo - Departmento de Matemáticas, Universidad Nacional de Colombia, Bogotá, Colombia (email)
Alfonso Castro - Department of Mathematics, Harvey Mudd College, Claremont, CA 91711, United States (email)
Arturo Sanjuán - Proyecto Curricular de Matemticas, Universidad Distrital Francisco José de Caldas, Bogotá, Colombia (email)

Abstract: We prove bifurcation at infinity for a semilinear wave equation depending on a parameter $\lambda$ 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 $L^\infty$ solutions tending to $+\infty$ 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.

Received: May 2016;      Revised: November 2016;      Available Online: December 2016.