Abstract: In this talk I will present some recent results on the Kirchhoff equation of nonlinear elasticity, describing transversal oscillations of strings and plates, with periodic boundary conditions. We are interested in the long-time existence and long-time dynamics of small amplitude solutions; to investigate such questions, we compute the normal form of the equation close to the elliptic equilibrium corresponding to the null solution. At the first step of normal form, one is able to erase from the equation all the cubic terms giving a nonzero contribution to the time evolution of the Sobolev norm of solutions; thus we deduce that, for initial data of size ϵ in Sobolev class, the time of existence of the solution is at least of order ϵ−4 (which improves the lower bound ϵ−2 coming from the linear theory). After the second step of normal form, there remain some resonant terms (which cannot be erased) of degree five that give a non-trivial contribution to the time evolution of the Sobolev norm of solutions. Nonetheless, we show that small initial data satisfying a suitable nonresonance condition produce solutions that exist over a time of order at least ϵ−6. On the other hand, we use the effective terms of degree five to construct some special solutions exhibiting a chaotic-like behavior. These results were obtained in collaboration with P. Baldi, F. Giuliani, M. Guardia.

This seminar is part of the activities of the Excellence Department Project CUP B83C23001390001.