  We consider the Cauchy problems in the whole space for wave equations with
higher derivative terms. We derive sharp growth estimates of the $L^2$-norm of
the solution itself in the case of the space 1, 2 dimensions. By imposing the
weighted $L^1$-initial velocity, we can get the lower and upper bound estimates
of the solution itself. In three or more dimensions, we observe that the
$L^2$-growth behavior of the solution never occurs in the ($L^2 \cap
L^1$)-framework of the initial data.
