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 (C1s+C2)−2 . As applications, we give: an estimate of order ɛ−4 for the lifespan Tɛ of the solution of the Cauchy problem with initial data of size ɛ in Sobolev spaces when the nonlinearity is given by any C2 function m ( s ) > 0 ; a necessary and sufficient condition for boundedness of a second order energy of the solutions.