Notes on a paper of Pokhozhaev

Abstract

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 $\varepsilon^{-4}$ for the lifespan $T_\varepsilon$ of the solution of the Cauchy problem with initial data of size $\varepsilon$ 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.