  We consider the semilinear wave equation in higher dimensions with power
nonlinearity in the super-conformal range, and its perturbations with lower
order terms, including the Klein-Gordon equation. We improve the upper bounds
on blow-up solutions previously obtained by Killip, Stovall and Vi\c{s}an [6].
Our proof uses the similarity variables' setting. We consider the equation in
that setting as a perturbation of the conformal case, and we handle the extra
terms thanks to the ideas we already developed in [5] for perturbations of the
pure power case with lower order terms.
