Polynomial Stability of Non-Linearly Damped Contraction Semigroups
Lassi Paunonen, David Seifert
TL;DR
Problem: establish polynomial decay for non-linearly damped abstract C0-semigroups of the form $\dot x=Ax-B\phi(B^* x)$ under a non-uniform observability estimate. Approach: prove a main theorem showing mild solutions converge to zero and classical solutions decay as $\|x(t)\|=O(t^{-1/(2\beta)})$, with improved rates $O(t^{-1/\alpha})$ when the resolvent satisfies $\|(is-A_B)^{-1}\|\lesssim 1+|s|^\alpha$ and $\phi$ is near linear near the origin; derive these via energy-dissipation inequalities and a nonlinear observability-resolvent link, including a wavepacket-to-observability argument under a spectral-gap scenario. Contributions: explicit rate formulas, a sharp refinement mechanism, and concrete instantiations for a 1D wave equation with weak damping and the SCOLE beam with a tip mass, under verifiable coefficient decay and eigenstructure assumptions. Significance: furnishes a practical, structurally transparent route to obtain decay rates for nonlinearly damped hyperbolic systems and guides verification via resolvent estimates and observability properties; yields explicit decay rates in classical PDE models.
Abstract
We investigate the stability properties of an abstract class of semi-linear systems. Our main result establishes rational rates of decay for classical solutions assuming a certain non-uniform observability estimate for the linear part and suitable conditions on the non-linearity. We illustrate the strength of our abstract results by applying them to a one-dimensional wave equation with weak non-linear damping and to an Euler-Bernoulli beam with a tip mass subject to non-linear damping.