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

Technical Summary: Global Existence for Wave and Beam Equations with Double Damping and a New Power Nonlinearity

Problem Formulation
The paper addresses the Cauchy problem in $\mathbb{R}^{n}$ for both wave and beam equations. The governing linear models incorporate a dual damping mechanism consisting of frictional damping and viscoelastic damping. The primary objective is to analyze the long-time behavior of solutions to these linear systems and to extend these findings to a nonlinear setting involving a specific power nonlinearity.

Central to the analysis is the introduction of a specific auxiliary quantity, defined for a solution $u$ as:
$$Q[u](t):=\|u_{t}(t,\cdot)+(-\Delta)^{\sigma}u(t,\cdot)\|_{L^{2}(\mathbb{R}^{n})}.$$
This quantity combines the time derivative of the solution with a fractional Laplacian term, serving as a critical metric for the system's decay properties.

Methodology and Linear Analysis
The authors first investigate the linear model (with a vanishing right-hand side). By analyzing the interaction between the frictional and viscoelastic damping terms, they establish that this specific combination of dissipative mechanisms induces an exponential decay of the quantity $Q[u](t)$ as $t \to \infty$. This decay behavior is distinct from the standard decay of the solution or its total energy alone, highlighting the unique stabilizing effect of the coupled damping terms on the specific structure of $Q[u](t)$.

Nonlinear Extension and Key Contributions
Motivated by the exponential decay observed in the linear case, the authors define a new power nonlinearity of the form:
$$N[u]:=|u_{t}+(-\Delta)^{\sigma}u|^{p}.$$
The core contribution of the work is the rigorous analysis of the nonlinear Cauchy problem where this term acts as the source or reaction term. The authors prove the existence of a unique global solution for any power $p > 1$.

A significant finding is that the nonlinearity $N[u]$ behaves as a "small perturbation" relative to the linear dynamics, despite the nonlinearity being present. Specifically, the authors demonstrate that the decay estimates for the unique global solution, the total energy, and the auxiliary quantity $Q[u](t)$ in the nonlinear case coincide exactly with the decay estimates derived for the corresponding linear Cauchy problem.

Results and Significance
The paper establishes that the interaction between frictional and viscoelastic damping is robust enough to maintain global existence and specific decay rates even when the system is subjected to the proposed power nonlinearity. The significance of the work lies in:

1. Defining a New Nonlinearity: The introduction of $N[u]$ based on the structure of the decaying linear quantity $Q[u](t)$.
2. Perturbation Stability: The surprising result that this nonlinearity does not alter the asymptotic decay profile of the solution, energy, or the auxiliary quantity $Q[u](t)$ for any $p > 1$.
3. Unified Decay: The proof that the nonlinear system retains the same decay estimates as the linear system with vanishing source, effectively showing that the double damping mechanism dominates the dynamics of the new nonlinearity.

The authors maintain a modest scope, focusing strictly on the mathematical derivation of global existence and the coincidence of decay estimates between the linear and nonlinear models, without positing broader physical applications or future experimental directions beyond the scope of the Cauchy problem in $\mathbb{R}^{n}$.