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

1. Problem Statement

The paper addresses the asymptotic stability of evolution equations governed by nonlinear maximal monotone operators in Hilbert spaces.

• The Linear Benchmark: In linear theory, the Borichev–Tomilov theorem (2010) provides a sharp characterization of polynomial decay rates for $C_0$-semigroups. It states that the energy decay rate $\|e^{-tA}A^{-1}\| \leq C t^{-1/\alpha}$ is equivalent to the growth of the resolvent on the imaginary axis: $\|(isI + A)^{-1}\| \leq C |s|^\alpha$ as $|s| \to \infty$.
• The Nonlinear Challenge: For nonlinear operators (e.g., $A(u) = -\Delta u + |u|^{p-1}u$ or nonlocal damping), the concept of a "spectrum" and the "imaginary axis" ($i\mathbb{R}$) are undefined. Consequently, the standard frequency-domain analysis used in the Borichev–Tomilov theorem is inapplicable.
• The Gap: There is a lack of a unified framework to determine polynomial decay rates for nonlinear, degenerate dissipative systems, particularly those where classical multiplier methods fail due to insufficient regularity of weak solutions.

2. Methodology: The Real Resolvent Framework

The authors propose a conceptual shift from the frequency domain to the regularization domain (real axis). Instead of analyzing the resolvent at high frequencies ($i\omega$), they analyze the behavior of the real resolvent equation near the origin ($\lambda \to 0^+$).

Core Equation:
They study the stationary problem associated with the resolvent step:
$$\lambda x_\lambda + A(x_\lambda) \ni y, \quad \lambda > 0$$
where $A$ is a maximal monotone operator and $y$ is a fixed source term.

The Nonlinear Tauberian Principle:
The central thesis is that the blow-up rate of the norm $\|x_\lambda\|$ as $\lambda \to 0^+$ reveals the effective nonlinear scaling (homogeneity) of the operator. This scaling dictates the polynomial decay rate of the associated semigroup $S(t) = e^{-tA}$.

• Linear Analogy: High-frequency resolvent growth $\leftrightarrow$ Slow decay.
• Nonlinear Analogy: Low-frequency (near-zero) real resolvent blow-up $\leftrightarrow$ Slow polynomial decay.

Key Technical Steps:

1. Homogeneity Analysis: For $\alpha$-homogeneous operators (where $A(\lambda u) = \lambda^\alpha A(u)$), the authors derive that if $\|x_\lambda\| \sim \lambda^{-\frac{1}{\alpha-1}}$, the energy decays as $E(t) \sim t^{-\frac{2}{\alpha-1}}$.
2. Coercive Dissipation: The method relies on establishing a coercive inequality of the form $\langle \xi, u \rangle \geq m \|u\|^{\alpha+1}$ for $\xi \in A(u)$. This leads to a differential inequality $E'(t) + c E(t)^{\frac{\alpha+1}{2}} \leq 0$, which yields the decay rate.
3. Handling Degeneracy: The framework distinguishes between "dissipative alignment" (where degeneracy occurs in the dissipated variable, allowing the resolvent method to work) and "misalignment" (where degeneracy affects the elastic part, requiring geometric multipliers).

3. Key Contributions

A. Theoretical Innovation

• Nonlinear Tauberian Principle: The paper establishes a rigorous abstract criterion (Theorems 3.2 and 3.3) linking the growth of the real resolvent $\|(\lambda I + A)^{-1}\|$ to polynomial decay rates, effectively creating a nonlinear counterpart to the Borichev–Tomilov theorem.
• Weak Solution Validity: Unlike classical multiplier methods which often require high regularity (e.g., $H^2$ solutions) to justify integration by parts, this resolvent-based approach allows for the derivation of decay rates for weak (mild) solutions via density arguments.

B. Diagnostic Capability

The framework acts as a diagnostic tool:

• If the resolvent remains bounded as $\lambda \to 0^+$, the system may exhibit exponential stability (if linear) or specific polynomial rates based on nonlinearity.
• If the resolvent blows up ($\|x_\lambda\| \to \infty$), it quantifies the "flatness" of the nonlinearity and predicts the exact algebraic decay rate.
• It identifies structural obstructions: If the degeneracy is in the "wrong" component (e.g., Kirchhoff-type damping where the coefficient depends on potential energy but damping acts on velocity), the resolvent remains bounded, and the method correctly signals that geometric/time-domain arguments are necessary.

4. Results and Applications

The authors validate their theory through ten diverse applications, covering wave equations, parabolic equations, and memory systems:

1. Nonlocal Kelvin–Voigt Damping (Applications I, II, V):
   • Model: Wave equation with damping depending on $\|\nabla u_t\|$.
   • Result: The resolvent grows as $\lambda^{-1}$. The method recovers the optimal decay rate $E(t) \sim (1+t)^{-1}$ for weak solutions, bypassing the regularity issues of classical multipliers.
2. Kirchhoff-Type Damping (Application IV):
   • Model: Damping coefficient depends on $\|\nabla u\|^2$ (elastic energy).
   • Result: The resolvent is bounded (no blow-up) because the degeneracy is in the elastic part, not the dissipative part. The authors demonstrate that the resolvent method correctly identifies this as a case requiring geometric control conditions (GCC) and time-domain multipliers, serving as a structural counterexample to the direct applicability of the Tauberian principle.
3. p-Laplacian Wave and Parabolic Equations (Applications VI, VII):
   • Model: Wave equation with $p$-Laplacian stiffness ($p>2$) and strong linear damping.
   • Result: The resolvent growth is fractional ($\lambda^{-1/p}$). This correctly predicts the decay rate $E(t) \sim t^{-p/(p-2)}$, capturing the "soft spring" phenomenon where the bottleneck is the degenerate stiffness, not the damping.
4. Degenerate Parabolic Equations (Applications VIII, IX):
   • Model: Diffusion with interior degeneracy ($a(x)=0$ on a set).
   • Result: The resolvent blows up as $\lambda^{-1}$, correctly diagnosing the failure of uniform decay due to invariant subspaces (insulating barriers). Conversely, with "complementary control" (reaction term active where diffusion is zero), the resolvent is bounded, and polynomial decay is recovered.
5. State-Dependent Memory (Application X):
   • Model: Viscoelastic wave equation with memory feedback modulated by the state.
   • Result: The memory provides observability (bounded resolvent for the linear part), while the state-dependent weight dictates the polynomial rate via the nonlinear Tauberian argument.

5. Significance and Impact

• Unification: The paper unifies a wide array of scattered results in the literature under a single "Nonlinear Tauberian Principle," providing a systematic way to analyze stability without reinventing the wheel for each specific PDE.
• Overcoming Regularity Barriers: It offers a robust alternative to the "multiplier method," which often fails for weak solutions in nonlinear settings. By shifting the analysis to the static resolvent equation, it avoids the need for high-order spatial derivatives in the energy estimates.
• Structural Insight: It clarifies the physical mechanisms of decay. It demonstrates that polynomial decay is not a spectral property but a manifestation of nonlinear coercivity and the specific scaling of the operator near equilibrium.
• Future Directions: The framework is presented as a foundational tool for analyzing complex systems like Timoshenko beams, Bresse systems, and mixed wave-plate structures, particularly those involving degenerate or nonlocal damping.

In summary, the paper successfully bridges the gap between linear spectral theory and nonlinear asymptotic analysis, providing a powerful, rigorous, and practical tool for determining polynomial stability in degenerate dissipative systems.