Multiplicity and regularity of large periodic solutions with rational frequency for a class of semilinear monotone wave equations

Fokam, Jean-Marcel

doi:10.1090/proc/12760

Multiplicity and regularity of large periodic solutions with rational frequency for a class of semilinear monotone wave equations
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by Jean-Marcel Fokam

Proc. Amer. Math. Soc. 145 (2017), 4283-4297

DOI: https://doi.org/10.1090/proc/12760

Published electronically: July 10, 2017

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Abstract:

We prove the existence of infinitely many classical large periodic solutions for a class of semilinear wave equations with periodic boundary conditions: \[ u_{tt}-u_{xx}+f(x,u)=0, \] \[ u(0,t)=u(\pi ,t) , u_x(0,t)=u_x(\pi ,t). \] Our argument relies on some new estimates for the linear problem with periodic boundary conditions, the Hausdorff-Young theorem of harmonic analysis and a variational formulation due to Rabinowitz. We also develop a new approach to the regularity of the distributional solutions by differentiating the equations and employing Gagliardo-Nirenberg estimates.

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