We consider the Cauchy problem for the nonlinear wave equation uttββΞxβu+q(t,x)u+u3=0 with smooth potential q(t,x)β₯0 having
compact support with respect to x. The linear equation without the nonlinear
term u3 and potential periodic in t may have solutions with exponentially
increasing as tββ norm H1(Rx3β). In [2] it was
established that adding the nonlinear term u3 the H1(Rx3β)
norm of the solution is polynomially bounded for every choice of q. In this
paper we show that Hk(Rx3β) norm of this global solution is also
polynomially bounded. To prove this we apply a different argument based on the
analysis of a sequence {Ykβ(nΟkβ)}n=0ββ with suitably
defined energy norm Ykβ(t) and $0 < \tau_k <1.