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

TL;DR

This paper proves global existence and exponential decay for wave and beam equations with combined damping and a novel power nonlinearity, showing that the nonlinearity acts as a small perturbation under certain conditions.

Contribution

It introduces a new power nonlinearity based on a combined damping term and demonstrates its perturbative effect on the decay properties of solutions.

Findings

Exponential decay of a new quantity Q[u](t) due to damping interaction.

The new nonlinearity N[u] is a small perturbation for p>1.

Decay estimates match those of the linear problem with zero right-hand side.

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