19-41 Massimiliano Berti, Philippe Bolle

Quasi-periodic solutions of nonlinear wave equations on Td (3934K, PDF) Jun 20, 19

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Abstract. We consider autonomous nonlinear wave equations (NLW) \[ u_{tt} −\Delta +V(x)u+g(x,u)=0, x \in \mathbb{T}^d \equiv \mathbb{R}^d/(2\pi\mathbb{Z})d \] in any space dimension $d \ge 1$, where $ V(x) \in C^\infty( \mathbb{T}^d, \mathbb{R})$ is a real valued multiplicative potential and the nonlinearity $g \in C^\infty( \mathbb{T}^d imes \mathbb{R}, \mathbb{R})$ has the form $g(x, u) = a(x)u^3 + O(u^4)$ with $ a(x)\in C^\infty( \mathbb{T}^d, \mathbb{R})$ We require that −\Delta +V(x) > eta Id, eta > 0. This condition is satisfied, in particular, if the potential V (x) \ge 0 and V (x) ot\equiv ≡ 0. In this Monograph we prove the existence of small amplitude time quasi-periodic solutions of (NLW).

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