  We are interested in almost global existence cases in the general theory for
nonlinear wave equations, which are caused by critical exponents of nonlinear
terms. Such situations can be found in only three cases in the theory, cubic
terms in two space dimensions, quadratic terms in three space dimesions and
quadratic terms including a square of unknown functions itself in four space
dimensions. Except for the last case, criterions to classify nonlinear terms
into the almost global, or global existence case, are well-studied and known to
be so-called null condition and non-positive condition.
  Our motivation of this work is to find such a kind of the criterion in four
space dimensions. In our previous paper, an example of the non-single term for
the almost global existence case is introduced. In this paper, we show an
example of the global existence case. These two examples have nonlinear
integral terms which are closely related to derivative loss due to high
dimensions. But it may help us to describe the final form of the criterion.
