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.

Computing the first step of quasilinear normal form, we 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 small initial data of size $\varepsilon$ in Sobolev class, the time of existence of the solution is at least of order $\varepsilon^{-4}$ (which improves the lower bound $\varepsilon^{-2}$ 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; this could be interpreted as a sign of non-integrability of the equation. Nonetheless, we show that small initial data satisfying a suitable nonresonance condition produce solutions that exist over a time of order at least $\varepsilon^{-6}$. On the other hand, we use such effective terms of degree five to construct some special solutions exhibiting a chaotic-like behavior.

In a more recent work in progress, we also study the normal form of a special Kirchhoff-type equation, which is globally well-posed in time for initial data in Sobolev class.

These results were obtained in collaboration with P. Baldi, F. Giuliani, M. Guardia, S. Marrocco.