  In this paper, we study weakly coupled systems for semilinear wave equations
with distinct nonlinear memory terms in general forms, and the corresponding
single semilinear equation with general nonlinear memory terms. Thanks to
Banach's fixed point theorem, we prove local (in time) existence of solutions
with the $L^1$ assumption on the memory kernels. Then, blow-up results for
energy solutions are derived applying iteration methods associated with slicing
procedure. We investigate interactions on the blow-up conditions under
different decreasing assumptions on the memory term. Particularly, a new
threshold for the kernels on interplay effect is found. Additionally, we give
some applications of our results on semilinear wave equations and acoustic wave
equations.
