From Linear to Nonlinear: A Resolvente criterion for Polynomial Stability of Semigroups Generated by Monotone Operators

TL;DR

This paper introduces a nonlinear resolvent criterion for polynomial stability of semigroups generated by monotone operators, extending spectral analysis techniques to nonlinear settings.

Contribution

It develops a new framework replacing spectral analysis with asymptotic analysis of the real resolvent equation for nonlinear maximal monotone operators.

Findings

Reveals the nonlinear scaling of operators through resolvent blow-up rates.

Establishes a nonlinear Tauberian principle linking resolvent behavior to decay rates.

Recovers optimal decay rates for wave equations with nonlocal damping.

Abstract

The Borichev--Tomilov theorem \cite{BT2010} provides a sharp characterization of polynomial decay for linear $C_0$-semigroups in terms of resolvent growth along the imaginary axis. In the nonlinear setting, the absence of a spectral theory renders the imaginary-axis approach inapplicable. In this paper, we develop a new framework for nonlinear maximal monotone operators in Hilbert spaces by replacing spectral analysis on $i\mathbb{R}$ with the asymptotic analysis of the \textit{real resolvent equation} \[ \lambda x_\lambda + \mathcal{A}(x_\lambda) \ni y, \quad \lambda \to 0^+. \] We show that, for homogeneous operators (and suitable perturbations), the blow-up rate of $\|x_\lambda\|$ at the origin reveals the effective nonlinear scaling of the operator and determines the corresponding polynomial decay rate of the associated semigroup through a coercive dissipation mechanism. This provides a nonlinear Tauberian-type principle for a broad class of degenerate dissipative systems. The approach recovers, in particular, the optimal $1/t$ decay for the wave equation with nonlocal Kelvin--Voigt damping recently obtained by Cavalcanti et al.\ (2025), and allows one to justify decay estimates for weak solutions in situations where classical multiplier methods require higher regularity. It also clarifies the structural limitations of the method, identifying regimes where additional geometric or time-domain arguments are necessary.