  The long time behavior of a kind of fully magnetic effected nonlinear
piezoelectric beam with viscoelastic infinite memory is considered. The
well-posedness of this nonlinear coupled PDEs system is showed by mean of the
semigroup theories and Banach fixed point theorem. Based on frequency domain
analysis, it is proved that the corresponding coupled linear system can be
indirectly stabilized exponentially by only one viscoelastic infinite memory
term, which is located on one equation of these strongly coupled PDEs. Then the
exponential decay of the solution to the nonlinear coupled PDEs' system is
established by the energy estimation method under certain condition.
