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

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.