  We consider the Cauchy problem in $\mathbb{R}^{n}$ for wave and beam
equations with frictional, viscoelastic damping, and a new power nonlinearity.
In addition to the solution and its total energy, we define the following
quantity:
$$Q[u](t):=\|u_{t}(t,\cdot)+(-\Delta)^{\sigma}u(t,\cdot)\|_{L^{2}(\mathbb{R}^{n})}.$$
  Our aim is to show that the interaction between frictional and viscoelastic
damping in a linear model leads to an exponential decay of $Q[u](t)$ as $t\to
\infty$. This decay motivates us to define a new power nonlinearity of the form
$N[u]:=|u_{t}+(-\Delta)^{\sigma}u|^{p}$. Surprisingly, $N[u]$ can be considered
a small perturbation for any $p>1$, in the sense that, the decay estimates of
the unique global solution, the total energy and $Q[u](t)$ coincide with those
for solutions to the corresponding linear Cauchy problem with vanishing
right-hand side.
