A Kirchhoff equation with infinite conservation laws

We prove here the existence of time-invariant functionals of all orders $k\ge 3$ for the solutions of the quasilinear Kirchhoff-Pokhozhaev equation $u_{tt}-( a\int_{\R^n} |\nabla u |^2 dx + b )^{-2 } \Delta u = 0$ with $a\ne 0$. Assuming also $b \ne 0$, we show that if $u$ is a suitably small smooth global solution of the Kirchhoff-Pokhozhaev equation, then the $L^2$ norms of all its derivatives remain uniformly bounded.