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

Abstract

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: $$Q[u](t):=\Vert u_t(t,\cdot )+(-\Delta {)}^{\sigma }u(t,\cdot ){\Vert }_{L^2({\mathbb{R}}^n)}.$$ 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.

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