Global existence for wave and beam equations with double damping and a new power nonlinearity

We consider the Cauchy problem in Rⁿ 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):=\|uₜ(t,·)+(-Δ)σu(t,·)\|_L²(Rⁿ). 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→ ∞. This decay motivates us to define a new power nonlinearity of the form N[u]:=|uₜ+(-Δ)σu|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.