Rational-Kernel Fractional Evolution Equations with Almost Sectorial Operators: A Resolvent Framework Unifying ABC and W Dynamics

TL;DR

We address fractional evolution equations driven by rational nonsingular kernels, notably ABC and W, for $0<\alpha<1$, in the setting of almost sectorial generators. We construct a unified $K$-resolvent framework using a Laplace multiplier $K(s)$ and a contour integral, proving well-posedness and a variation-of-constants formula, with sharp fractional smoothing $\|A^\gamma V_K(t)\| \le C_\gamma t^{-\alpha\gamma}$ for $\gamma\in[0,1)$. The key geometric idea is the mapping $z=s^{\alpha-1}$, which redirects low-frequency Laplace modes to high spectral parameters, avoiding the origin where Caputo dynamics fail. Specializing to ABC ($\beta=1$) and W ($0<\beta\le 1$) dynamics, the theory applies to degenerate diffusions like Kimura and Bessel operators and yields both theoretical guarantees and numerical illustrations of the predicted smoothing and decay.

Abstract

We study fractional evolution equations driven by rational-kernel time operators with non-singular memory, including the Atangana-Baleanu-Caputo operator and a generalized W-operator. These operators are characterized by Laplace symbols that do not necessarily belong to the classical Bernstein class. The analysis is carried out in the framework of almost sectorial operators, which allows resolvent estimates beyond standard analytic semigroup theory. Existence, uniqueness, and temporal regularity of mild solutions are established by Laplace transform techniques and contour integration, leading to the construction of associated resolvent families. A unified resolvent framework is developed, enabling a precise comparison between ABC and W dynamics and clarifying the influence of rational memory kernels on decay and smoothing properties. Several examples, including fractional diffusion-type equations, illustrate the abstract theory and highlight the impact of non-singular memory on long-time behavior.

Figures (5)

• Figure 1: Geometric redirection induced by the mapping $z=s^{\alpha-1}$.
• Figure 2: Decay of $\|K^\gamma V_{\mathrm{ABC}}(t)u_0\|$ for the Kimura operator with ABC dynamics ($\alpha=0.5$, $\gamma=0.5$). The dashed line corresponds to the theoretical bound $t^{-\alpha\gamma}$, while the dotted line shows the faster reference rate $t^{-\gamma}$.
• Figure 3: Local decay exponent $-\frac{d\log \|K^\gamma V_{\mathrm{ABC}}(t)u_0\|}{d\log t}$ for the Kimura operator. The horizontal dashed and dotted lines correspond to $\alpha\gamma$ and $\gamma$, respectively.
• Figure 4: Decay of $\|\mathcal{B}_\nu^\gamma u(t)\|$ for the Bessel operator ($\nu=0.25$) with W dynamics ($\alpha=0.5$, $\beta=0.8$, $\gamma=0.5$). The dashed and dotted lines correspond to the reference rates $t^{-\alpha\gamma}$ and $t^{-\gamma}$, respectively.
• Figure 5: Local decay exponent $-\frac{d\log \|\mathcal{B}_\nu^\gamma u(t)\|}{d\log t}$ for the Bessel operator with W dynamics. The horizontal reference lines correspond to $\alpha\gamma$ and $\gamma$.

Theorems & Definitions (45)

• Definition 2.1: Almost sectorial operator
• Remark 2.2: Interpretation and scope
• Remark 2.3
• Proposition 3.1: Structural failure of Caputo dynamics
• proof
• Remark 3.2
• Definition 4.1: Admissible kernel multiplier
• Remark 4.2
• Lemma 4.3: Geometric redirection
• proof