Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Marcelo M. Cavalcanti, Valéria N. Domingos Cavalcanti, Jaime E. Munõz Rivera

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.

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 -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 with the asymptotic analysis of the \textit{real resolvent equation} We show that, for homogeneous operators (and suitable perturbations), the blow-up rate of 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 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.

Paper Structure

This paper contains 78 sections, 12 theorems, 223 equations.

Table of Contents

  1. Introduction: From Linear to Nonlinear Stability
  2. The Linear Benchmark: Borichev--Tomilov
  3. The Challenge
  4. Our Proposal
  5. Dissipative alignment.
  6. Preliminaries and The Resolvent Framework
  7. Maximal Monotone Operators
  8. Interpretation of the Linear Benchmark
  9. The Nonlinear Challenge and The Real Resolvent Proposal
  10. Heuristic Scaling Argument
  11. Homogeneity and Assumptions
  12. Heuristic Motivation and Proof Strategy
  13. Rigorous Proof of Theorem \ref{['thm:main']}
  14. A rigorous decay principle for homogeneous maximal monotone operators
  15. A General Decay Lemma
  16. ...and 63 more sections

Key Result

Theorem 3.2

Let $A$ be an $\alpha$-homogeneous maximal monotone operator with $\alpha > 1$. Assume that the solution $x_\lambda$ to the real resolvent equation satisfies the growth estimate as $\lambda \to 0^+$: Revised Statement: Let $u(t)$ be the solution to the evolution problem $\dot{u} + Au \ni 0$. The following statements are formally linked through the scaling structure of the operator:: $\blacktria

Theorems & Definitions (49)

  • Remark 2.1: Comparison of Mechanisms
  • Definition 3.1: $\alpha$-Homogeneity
  • Theorem 3.2: Nonlinear Resolvent Criterion
  • proof
  • Theorem 3.3: Abstract polynomial decay criterion
  • proof
  • Remark 3.4: Weak solutions
  • Corollary 3.5: Homogeneous case
  • proof
  • Remark 3.6: How the applications fit
  • ...and 39 more