Notes on a paper of Pokhozhaev

We prove a second order identity for the Kirchhoff equation which yields, in particular, a simple and direct proof of Pokhozhaev’s second order conservation law when the nonlinearity has the special form $(C_{1}s+C_2{)}^{-2}$. As applications, we give: an estimate of order ${\epsilon }^{-4}$ for the lifespan $T_{\epsilon }$ of the solution of the Cauchy problem with initial data of size $\epsilon$ in Sobolev spaces when the nonlinearity is given by any $C^2$ function $m(s)>0$; a necessary and sufficient condition for boundedness of a second order energy of the solutions.