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 . As applications, we give: an estimate of order ɛ −4 for the lifespan T ɛ of the solution of the Cauchy problem with initial data of size ɛ in Sobolev spaces when the nonlinearity is given by any C 2 function ; a necessary and sufficient condition for boundedness of a second order energy of the solutions.
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