Abstract
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 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
m
(
s
)
>
0
; a necessary and sufficient condition for boundedness of a second order energy of the solutions.
Subject
Applied Mathematics,General Physics and Astronomy,Mathematical Physics,Statistical and Nonlinear Physics
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