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} [ λx_λ+ \mathcal{A}(x_λ) \ni y, \quad λ\to 0^+. ] We show that, for homogeneous operators (and suitable perturbations), the blow-up rate of $|x_λ|$ 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.

• The paper extends the Borichev–Tomilov theorem to nonlinear maximal monotone operators by using a resolvent blow-up characterization to link operator homogeneity with decay rates.
• It establishes a nonlinear Tauberian principle where the asymptotic growth of the real resolvent determines the precise polynomial decay rate of the semigroup energy.
• The framework is validated on various PDE models, including nonlocal Kelvin–Voigt damping and degenerate parabolic equations, proving its robustness and broad applicability.

Nonlinear Resolvent Growth and Polynomial Stability of Semigroups: From Spectral to Dissipation Structures

Overview

This work introduces a formal nonlinear analogue of the Borichev–Tomilov (BT) theorem for polynomial stability of $C_0$-semigroups, extending the resolvent-based spectral criterion from the linear to the nonlinear (maximal monotone operator) context. The absence of a well-defined spectrum for nonlinear operators precludes frequency-domain analysis; the authors instead develop a resolvent blow-up characterization grounded in the asymptotic growth of solutions to the real resolvent equation as the parameter $\lambda\to 0^+$. This framework yields a nonlinear Tauberian principle, linking operator homogeneity and dissipation to precise polynomial decay rates for general classes of nonlocal, degenerate, and nonlinear dissipative systems.

Nonlinear Resolvent Framework and Tauberian Principle

The main conceptual shift is from the analysis of resolvent norms along the imaginary axis, $\| (isI + A)^{-1} \|$, which governs linear decay rates, to the examination of the real resolvent equation

$$\lambda x_\lambda + \mathcal{A}(x_\lambda) \ni y, \qquad \lambda \to 0^+$$

for maximal monotone operators $\mathcal{A}$ in a Hilbert space. The core insight is that the asymptotic scaling (blow-up) of $\|x_\lambda\|$ as $\lambda \downarrow 0$ encodes the effective degeneracy of the operator near equilibrium, which directly determines the system’s long-term decay.

The authors formalize this by showing that if the blow-up profile

$\|x_\lambda\| \lesssim \lambda^{-1/(\alpha-1)}$

prevails for an $\alpha$-homogeneous maximal monotone operator ($A(\lambda x) = \lambda^\alpha A(x)$), then the associated semigroup admits an optimal polynomial rate of energy decay

$\lambda\to 0^+$0

where $\lambda\to 0^+$1 denotes the physical energy, and the exponent matches the scaling $\lambda\to 0^+$2 of the nonlinearity.

This nonlinear Tauberian criterion generalizes the BT theorem, which in the linear case connects the resolvent growth along $\lambda\to 0^+$3 to decay via $\lambda\to 0^+$4 iff $\lambda\to 0^+$5, but is formulated at the level of the real resolvent for the nonlinear maximal monotone setting.

Main Structural Principles and Results

The paper rigorously establishes the following:

• Resolvent Growth–Decay Correspondence: For homogeneous (and suitable perturbations of) maximal monotone operators, the growth rate of the solution to $\lambda\to 0^+$6 as $\lambda\to 0^+$7 is both necessary and sufficient for deducing a precise polynomial decay rate of the semigroup. This offers a direct replacement for spectral localization in the absence of linear structure.
• Coercivity and Dissipative Alignment: Decay ensues when the degeneracy of the nonlinearity coincides with the variable through which dissipation acts; misalignment results in a loss of resolvent control and necessitates additional geometric or time-domain arguments.
• Nonlinear Differential Energy Inequality: The decaying energy satisfies a nonlinear ODE

$\lambda\to 0^+$8

whose solution precisely recapitulates the predicted polynomial decay rate. This is achieved via direct Lyapunov methods, bypassing spectral/multiplier obstacles.

• Robustness to Weak Data: The framework enables decay estimates for weak (mild) solutions, avoiding excessive regularity demands typical in the classical multiplier method.

Exemplary Applications

The abstract theory is corroborated by an extensive spectrum of PDE models representing degenerate and nonlocal dissipative mechanisms typical in wave and parabolic equations. Notable cases include:

• Nonlocal Kelvin–Voigt Damping: For the wave equation with energy-dependent or nonlocal feedback, e.g.,

$\lambda\to 0^+$9

the resolvent growth is shown to scale as $\| (isI + A)^{-1} \|$0, leading to an optimal decay $\| (isI + A)^{-1} \|$1, matching the sharp results previously established by direct means.

• Generalized Nonlocal Nonlinear Damping: Analysis extends to equations where the damping is governed by nonlinear state-dependent coefficients and nonlocal functionals, with homogeneity directly controlling the decay exponent.
• Parabolic and Memory-Type Systems: For degenerate parabolic equations (e.g., weighted $\| (isI + A)^{-1} \|$2-Laplacian or interior degeneracy), the resolvent growth precisely diagnoses whether uniform (or even any) decay is possible. For viscoelastic equations with nonlinear memory, the limiting decay is shown to be dictated by the nonlinear weight in the memory kernel, with the resolvent index quantifying the rate.
• Kirchhoff-Type Models and Limitations: In classes where the degeneracy lies in an undamped component, the resolvent remains bounded and fails to capture decay, necessitating geometric control or multiplier-based observability. This sharpens the boundaries of applicability for the nonlinear resolvent theory.

Main Technical Outcomes

Theorem: Nonlinear Resolvent Criterion for Polynomial Decay

Given an $\| (isI + A)^{-1} \|$3-homogeneous maximal monotone operator $\| (isI + A)^{-1} \|$4 and real resolvent scaling $\| (isI + A)^{-1} \|$5 as $\| (isI + A)^{-1} \|$6, solutions to $\| (isI + A)^{-1} \|$7 satisfy

$\| (isI + A)^{-1} \|$8

Abstract Polynomial Decay via Coercivity

If

$\| (isI + A)^{-1} \|$9

holds for all $$\lambda x_\lambda + \mathcal{A}(x_\lambda) \ni y, \qquad \lambda \to 0^+$$0, $$\lambda x_\lambda + \mathcal{A}(x_\lambda) \ni y, \qquad \lambda \to 0^+$$1, then

$$\lambda x_\lambda + \mathcal{A}(x_\lambda) \ni y, \qquad \lambda \to 0^+$$2

and thus

$$\lambda x_\lambda + \mathcal{A}(x_\lambda) \ni y, \qquad \lambda \to 0^+$$3

Theoretical and Practical Implications

• Extension of Linear-Spectral Paradigms: The real-resolvent blow-up provides a rigorous, quantitative counterpart to classical linear decay via resolvent growth, thus unlocking polynomial stability analysis for a much broader, nonlinear class of dissipative systems.
• Tool for Diagnosing Decay and Instability: Resolvent growth rates not only predict decay but, when divergent ($$\lambda x_\lambda + \mathcal{A}(x_\lambda) \ni y, \qquad \lambda \to 0^+$$4 scaling), diagnose the existence of obstruction to stabilization (e.g., invariant subspaces for interior degeneracy).
• Bypass of Strong Regularity and Compactness Requirements: The analysis obviates the need for strong solution frameworks or compact embedding techniques, particularly significant for weak or degenerate damping, or in settings where attractor theory is unavailable.
• Frameworks for Memory, Nonlocal, and Degenerate Problems: Viscoplastic/viscoelastic models with degenerate or state-dependent memory are handled uniformly, revealing that the main structural hurdle is nonlinear scaling, not the loss of spectral tools.

Future Directions

The methodology is amenable to generalizations toward mixed hyperbolic systems (Timoshenko, Bresse), systems with more intricate geometric or nonlocal dissipative structures, and further analysis of robustness to semilinear source perturbations where subcritical growth guarantees preservation of decay rates. Investigations into optimal control and stabilization protocols for such nonlinear PDEs are directly motivated by these findings.

Conclusion

This article provides a rigorous and unifying nonlinear Tauberian principle for the stability of semigroups generated by monotone operators, sitting as a conceptual successor to the Borichev–Tomilov theorem in the linear regime. Transforming the spectral resolvent condition into the asymptotic analysis of the real resolvent equation, the authors offer a robust tool for predicting and verifying polynomial decay rates in dissipative evolution equations well beyond the reach of multiplier, spectral, or contradiction methods. This supplies a central organizing principle for the stabilization theory of nonlinear and degenerate PDEs.

Reference: "From Linear to Nonlinear: A Resolvente criterion for Polynomial Stability of Semigroups Generated by Monotone Operators" (2604.03811)