| Abstract: |
| We consider Klein-Gordon equations with cubic nonlinearity in three spatial dimensions, which are Hamiltonian perturbations of the linear one with potential. It is assumed that the corresponding Klein-Gordon operator admits an arbitrary number of possibly degenerate eigenvalues in $(0, m)$, and hence the unperturbed linear equation has multiple time-periodic solutions known as bound states. In 1999, Soffer and Weinstein discovered a mechanism called Fermi`s Golden Rule for this nonlinear system in the case of one simple but relatively large eigenvalue $\Omega \in (m/3 , m)$, by which energy is transferred from discrete to continuum modes and the solution still decays in time. In our first work, we solved the general one simple eigenvalue case. In our second work, we solved this problem in full generality: multiple and simple or degenerate eigenvalues in $(0, m)$. Indeed, we obtained the sharp rate of energy transfer from one discrete state to continuum modes in the general case. The proof is based on a kind of pseudo-one-dimensional cancellation structure in each eigenspace, a renormalized damping mechanism, and an enhanced damping effect. It also relies on a refined Birkhoff normal form transformation and an accurate generalized Fermi`s Golden Rule building upon the results of Bambusi and Cuccagna. This is a joint work with Prof. Zhen Lei and Dr. Jie Liu. |
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