Mathematics

The study of existence and multiplicity of time-periodic solutions for semilinear Klein-Gordon equation has been recently of interest in general relativity as a toy model to understand stability properties of Anti-De Sitter spacetime under certain small perturbations. I will present a result on existence and multiplicity of time-periodic solutions, with time frequency satisfying strong Diophantyne conditions, for semilinear Klein-Gordon equation on $S^3$, where the nonlinear term is of the form $f(u)=u^p$, $(p=2, 3, 5)$.The proof of our result relies on the variational structure of the equation, in particular we perform a variational Lyapunov-Schmidt decomposition, which reduces to the search of critical points of a restricted Euler-Lagrange functional by a mountain pass-argument. Compactness properties of its gradient are obtained by developing new Strichartz-Type estimates for the solutions of linear Klein-Gordon equation on $S^3$. This is a joint work with M. Berti and B. Langella.