Research on the Existence and Uniqueness of Solutions to Fractional Diffusion Equations with Three-Parameter Damping Terms

Abstract

This paper investigates a fractional diffusion equation incorporating a three-parameter damping. By employing a generalized Mittag–Leffler function alongside the associated Riemann–Liouville resolvent family, we establish the well-posedness of strong solutions. This model extends the classical two-parameter undamped case, thereby ensuring consistency with the existing theoretical framework.

1. Introduction

In this paper, we focus on establishing the existence and uniqueness of solutions for fractional diffusion equations with three-parameter damping terms, specifically addressing the following Riemann–Liouville abstract fractional Cauchy problem ${\left(RLAFCP\right)}_{x_0,k}$:

$${\left(RLAFCP\right)}_{x_0,k}\left\{\begin{array}{cc}{D}_{\widehat{t}}^{\mu }u(\widehat{t},x)+e{D}_{\widehat{t}}^{\nu }u(\widehat{t},x)+k{D}_{\widehat{t}}^{\xi }u(\widehat{t},x)=Au(\widehat{t},x),\hfill & \widehat{t}>0,x\in \Omega \hfill \\ \underset{\widehat{t}\to 0^{+}}{lim}(G_{1-\mu }\ast u)(\widehat{t},x)=x_0\left(x\right),\hfill & x\in \Omega \hfill \end{array}\right.$$

The parameters satisfy the range $0<\xi <\nu <\mu \le 1$, The function $u(\widehat{t},x)$ denotes the state variable describing the diffusion process, where $\widehat{t}>0$ is the time variable and x $\in \Omega$ with $\Omega \subset {\mathbb{R}}^n$ being a unbounded spatial domain. The operator $A:D\left(A\right)\subset X\to X$ is a closed linear operator defined on a Banach space $X=L^2(\Omega )$, where the domain $D\left(A\right)$ is equipped with the graph norm ${\parallel ·\parallel }_{D\left(A\right)}=\parallel ·\parallel +\parallel A·\parallel$, this norm is essential for measuring the regularity of solutions within the operator domain. $x_0\left(x\right)$ represents the initial state. Here, $D_{\widehat{t}}^{\mu }$, $D_{\widehat{t}}^{\nu }$, and $D_{\widehat{t}}^{\xi }$ represent the Riemann–Liouville fractional derivative operators of orders $\mu$, $\nu$, and $\xi$, respectively. The function $G_{1-\mu }(\widehat{t},x)={\displaystyle \frac{{\widehat{t}}^{-\mu }}{\Gamma (1-\mu )}}$, where e denotes a real constant and $k\in \mathbb{R}$ is the coefficient of the additional damping term.

This model is a natural generalization of the pioneering work by Mei and Peng [1], who first systematically studied the two-parameter damped RLAFCP $D_{\widehat{t}}^{\mu }u\left(\widehat{t}\right)+e{D}_{\widehat{t}}^{\nu }u\left(\widehat{t}\right)=Au\left(\widehat{t}\right)$ by establishing the $(\alpha ,\beta ,c)$-resolvent family theory, the case $k=0$ corresponds to the RLAFCP with two damping terms. When $e=0$ and $k=0$, the system reduces to the classical RLAFCP [2]. For a comprehensive treatment of fractional calculus, we refer the reader to [3].

Fractional differential equations have become a central focus in applied mathematics and interdisciplinary research due to their ability to accurately model memory effects and nonlocal phenomena in physical, engineering, and biological systems [4,5]. Among these, the RLAFCP serves as a fundamental framework for describing complex dynamic behaviors such as viscoelastic vibrations and anomalous diffusion. The study of existence, uniqueness, and resolvent families for such equations continues to be a pivotal area in fractional functional analysis.

Current research on the RLAFCP primarily focuses on models with a single damping term, $D_{\widehat{t}}^{\mu }u\left(\widehat{t}\right)+e{D}_{\widehat{t}}^{\nu }u\left(\widehat{t}\right)=Au\left(\widehat{t}\right)$ with $0<\nu <\mu \le 1$. While such models effectively capture single-scale damping mechanisms, they have limitations in practical applications. For example, in multi-frequency viscoelastic systems, energy dissipation often exhibits multi-scale characteristics, involving high-frequency instantaneous damping as well as medium-frequency continuous damping. A single damping term is insufficient to fully characterize such complex dynamic behavior. Similarly, in fractional diffusion-reaction systems, the capacity of lower-order fractional derivatives to describe short-time diffusion coefficient perturbations falls outside the scope of existing two-parameter models [6,7]. This gap in modeling multi-scale effects highlights the need to extend the classical RLAFCP framework.

The incorporation of the third parameter $\xi$ is motivated by both physical realism and mathematical completeness: it specifically characterizes high-frequency or short-time damping effects, complementing $\nu$ that describes medium-scale damping and $\mu$ that describes low-scale damping, thereby comprehensively capturing the characteristics of multi-scale energy dissipation. Meanwhile, extending the resolvent family theory to the three-parameter case enriches the operator-theoretic framework for fractional evolution equations [8], and this generalization necessitates refined adjustments to both the Mittag–Leffler function and the fundamental properties of resolvent operators.

To address this gap, the present work introduces a significant generalization of the classical RLAFCP by incorporating an additional lower-order fractional damping term $k{D}_{\widehat{t}}^{\xi }u(\widehat{t},x)$, where $k\in \mathbb{R}$ is the damping coefficient and $0<\xi <\nu <\mu \le 1$. The resulting multi-parameter model, $D_{\widehat{t}}^{\mu }u(\widehat{t},x)+e{D}_{\widehat{t}}^{\nu }u(\widehat{t},x)+k{D}_{\widehat{t}}^{\xi }u(\widehat{t},x)=Au(\widehat{t},x)$ provides a more flexible framework for capturing complex dissipation mechanisms. Specifically, compared with the two-parameter model, our model achieves the precise separation of high-frequency, medium-frequency, and low-frequency three-scale damping for the first time by adding the parameter $\xi$; in contrast to Liu [6] multi-term numerical model, this study establishes a rigorous abstract operator theory, filling the theoretical gap of three-parameter damped RLAFCP. The main contributions of this paper are threefold:

1. We rigorously define the parameter range and physical interpretation of $\xi$, establishing its compatibility with the singularity of the original model at $\widehat{t}=0$.

2. We generalize the concept of an $(\mu ,\nu ,\xi ,e,k)$-resolvent family and characterize the solution structure through a three-parameter generalized Mittag–Leffler function [9,10].

3. We prove theorems on the existence and uniqueness of strong solutions based on the resolvent family theory [8,11], and demonstrate the applicability of the theory with examples involving viscoelastic vibrations and fractional diffusion processes.

Developing a well-posedness theory for the three-parameter fractional diffusion equation with damping requires a rigorous definition of strong solutions. A strong solution is defined as a function that satisfies the governing equation within the operator domain, complies with the prescribed initial conditions, and belongs to appropriate regularity classes. This foundational concept underpins the subsequent analysis of existence and uniqueness. We formally define a strong solution to the $(\mu ,\nu ,\xi ,e,k)-{RLAFCP}_{x_0,k}$ problem as follows:

Definition 1.

A function $u:(0,\infty )\times \Omega \to X$ is referred to as a strong solution to the $(\mu ,\nu ,\xi ,e,k)-RLAFC{P}_{x_0,k}$ if it satisfies the three core conditions below:

• (a) For any initial state $x_0\left(x\right)\in X$, the mapping $\widehat{t}\mapsto u(\widehat{t},x)$ is continuous on $(0,\infty )$ for each fixed $x\in \Omega$, and the convolutions of u with the Gamma kernels $G_{1-\mu }$, $G_{1-\nu }$, $G_{1-\xi }$ belong to the local Sobolev space $W_{loc}^{1,1}((0,\infty )\times \Omega ;X)$. This is formally expressed as:

  $$G_{1-\mu }\ast u(\widehat{t},x),G_{1-\nu }\ast u(\widehat{t},x),G_{1-\xi }\ast u(\widehat{t},x)\in W_{loc}^{1,1}((0,\infty )\times \Omega ;X)$$

  (1)

Note: All symbols ∗ in this paper denote the convolution with respect to the time variable $\widehat{t}$, and there is no convolution operation in the spatial dimension.

$W_{loc}^{1,1}((0,\infty )\times \Omega ;X)$ denotes the space of functions whose first-order weak derivatives are locally integrable over $(0,\infty )$. This condition is a crucial regularity requirement that ensures the Riemann–Liouville fractional derivatives of the function u are well-defined for the subsequent dynamic analysis.

• (b) The fractional differential equation with three-parameter damping holds pointwise in the Banach space $X=L^2(\Omega )$ for all $\widehat{t}>0$ and $x\in \Omega$:

  $$D_{\widehat{t}}^{\mu }u(\widehat{t},x)+e{D}_{\widehat{t}}^{\nu }u(\widehat{t},x)+k{D}_{\widehat{t}}^{\xi }u(\widehat{t},x)=Au(\widehat{t},x)$$

Here, $D_{\widehat{t}}^{\delta }$ denotes the Riemann–Liouville fractional derivative of order δ, $A:D\left(A\right)\subset X\to X$ is the closed linear operator defining the problem. The constants $e,k\in \mathbb{R}$ represent the damping coefficients. This condition guarantees that the function $u(\widehat{t},x)$ precisely satisfies the governing dynamic equation.

• (c) The initial condition is satisfied in the convolution sense for each $x\in \Omega$, where the highest-order fractional convolution limit specifies the initial state, and the lower-order limits are naturally determined by the parameter hierarchy:

  $$\underset{\widehat{t}\to 0^{+}}{lim}(G_{1-\mu }\ast u)(\widehat{t},x)=x_0\left(x\right),$$

  which implies the lower-order convolution limits vanish for all $x\in \Omega$:

  $$\underset{\widehat{t}\to 0^{+}}{lim}(G_{1-\nu }\ast u)(\widehat{t},x)=\underset{\widehat{t}\to 0^{+}}{lim}(G_{1-\xi }\ast u)(\widehat{t},x)=0$$

  (2)

This implication follows from the parameter ordering $0<\xi <\nu <\mu \le 1$ and the semigroup property of fractional integrals: the lower-order Gamma kernels $G_{1-\nu }$, $G_{1-\xi }$ correspond to weaker singularities at $\widehat{t}=0$.

This definition is generalized from the strong solution definition for two-parameter damped RLAFCP proposed by Mei and Peng [1]. The original work only considers the two-parameter damping terms $D_{\widehat{t}}^{\mu }u+e{D}_{\widehat{t}}^{\nu }u$, while this paper extends it to the three-parameter scenario by adding the lower-order damping term $k{D}_{\widehat{t}}^{\xi }u$. The core regularity requirements and convolution-type initial conditions remain consistent with the original work.

2. Preliminaries

This section introduces the fundamental concepts and mathematical tools necessary for the subsequent analysis, establishing a rigorous theoretical foundation for the study of three-parameter damped fractional diffusion equations. Let $A:D\left(A\right)\subset X\to X$ be a closed and densely defined linear operator on a Banach space X, where the domain $D\left(A\right)$ is equipped with the graph norm ${\parallel x\parallel }_{D\left(A\right)}=\parallel x\parallel +\parallel Ax\parallel$. The parameters satisfy the constraints $0<\xi <\nu <\mu \le 1$, with $e,k\in \mathbb{R}$ denoting damping coefficients that characterize multi-scale energy dissipation effects. The Gamma function is defined for $\delta >0$ as

$$\Gamma \left(\delta \right)={\int }_0^{\infty }{s}^{\delta -1}{e}^{-s}ds,\delta >0$$

This function obeys the recursive relation $\Gamma (\delta +1)=\delta \Gamma \left(\delta \right)$. Correspondingly, the Gamma kernel $G_{\delta }\left(\widehat{t}\right)$ is defined as follows:

$$G_{\delta }\left(\widehat{t}\right)=\left\{\begin{array}{cc}{\displaystyle \frac{{\widehat{t}}^{\delta -1}}{\Gamma \left(\delta \right)}},\hfill & \widehat{t}>0,\hfill \\ 0,\hfill & \widehat{t}\le 0\hfill \end{array}\right.$$

Notably, the Gamma kernel forms the core component of the fractional convolution operations in this study, and its piecewise definition ensures the well-posedness of the fractional integral operator $J_{\widehat{t}}^{\delta }$ on the domain $(0,\infty )$.

Definition 2.

For a function u belonging to the locally integrable function space $L_{loc}^1((0,\infty )\times \Omega ;X)$, the fractional integral operator $J_{\widehat{t}}^{\delta }$ of order $\delta >0$ is defined as the convolution of u with the Gamma kernel $G_{\delta }\left(\widehat{t}\right)$, specifically:

$$J_{\widehat{t}}^{\delta }u(\widehat{t},x)=G_{\delta }\ast u(\widehat{t},x)={\int }_0^{\widehat{t}}{\displaystyle \frac{{(\widehat{t}-\tau )}^{\delta -1}}{\Gamma \left(\delta \right)}}u(\tau ,x)d\tau ,x\in \Omega$$

A fundamental property of the operator $J_{\widehat{t}}^{\delta }$ is its semigroup property: for any positive real numbers $\delta >0$ and $\eta >0$, the composition of $J_{\widehat{t}}^{\delta }$ and $J_{\widehat{t}}^{\eta }$ equals the fractional integral operator of order $\delta +\eta$, i.e., $J_{\widehat{t}}^{\delta }{J}_{\widehat{t}}^{\eta }=J_{\widehat{t}}^{\delta +\eta }$, that is $\delta ,\eta >0$.

This is the standard definition of the Riemann–Liouville fractional integral operator, widely present in literature on fractional calculus and abstract fractional equations [3,5].

Definition 3.

Let u be a function belonging to the locally integrable function space $L_{loc}^1((0,T)\times \Omega ;X)$, with the fractional order satisfying $0<\delta <1$. The Riemann–Liouville fractional derivative operator $D_{\widehat{t}}^{\delta }$ of u with respect to the time variable $\widehat{t}$ is defined as follows: provided that the fractional integral $J_{\widehat{t}}^{1-\delta }$u belongs to the locally weakly differentiable space $W_{loc}^{1,1}((0,T)\times \Omega ;X)$, then

$$D_{\widehat{t}}^{\delta }u(\widehat{t},x)={\displaystyle \frac{d}{d\widehat{t}}}{J}_{\widehat{t}}^{1-\delta }u(\widehat{t},x)={\displaystyle \frac{d}{d\widehat{t}}}{\int }_0^{\widehat{t}}{\displaystyle \frac{{(\widehat{t}-\tau )}^{-\delta }}{\Gamma (1-\delta )}}u(\tau ,x)d\tau ,x\in \Omega$$

This is the classic form of the Riemann–Liouville fractional derivative for $0<\delta <1$, first proposed by Riemann and Liouville, and later widely applied to abstract fractional Cauchy problems [2,3]. This integral is well-defined, as $G_{\delta }\left(\widehat{t}\right)$ is locally integrable [3] and u is continuous and bounded per Definition 1, which ensures the integrability of their product.

Definition 4.

The three-parameter Mittag–Leffler function is defined by the following:

$$E_{\mu -\nu ,\mu -\xi ,\mu }(z,w)=\sum _{k=0}^{\infty }\sum _{m=0}^{\infty }{\displaystyle \frac{z^k{w}^m}{\Gamma \left(\right(\mu -\nu )k+(\mu -\xi )m+\mu )k!m!}}$$

(3)

This function is an extension of the two-parameter generalized Mittag–Leffler function proposed by Borah and Nath [10] and Gomez and Kilicman [12].

Remark 1.

The three-parameter generalized Mittag–Leffler function $E_{\mu -\nu ,\mu -\xi ,\mu }(z,w)$ demonstrates strong compatibility with the classical single-parameter Mittag–Leffler function, as evidenced by two typical degeneracy cases:

• 1. When the fractional orders satisfy $\nu =\mu$ and $\xi =\mu$ simultaneously, the double-index summation in the three-parameter function simplifies significantly. Since $\mu -\nu =0$ and $\mu -\xi =0$, the function degenerates into the classical single-parameter Mittag–Leffler form:

  $$E_{\mu -\mu ,\mu -\mu ,\mu }(z,w)=E_{\mu }(z+w)$$
• 2. When the constraint $\xi =\mu$ holds, the term $\mu -\xi =0$ eliminates the summation over the parameter w in the three-parameter function. Consequently, the function reduces to the classical two-parameter Mittag–Leffler function:

  $$E_{\mu -\nu ,\mu -\mu ,\mu }(z,w)=E_{\mu -\nu ,\mu }\left(z\right)$$

Remark 2.

The three-parameter Mittag–Leffler function is a bivariate extension of the classical Mittag–Leffler function and its generalization by Prabhakar. This extension enhances the flexibility of the Mittag–Leffler family by introducing a double-index summation, making it particularly well suited to modeling complex multi-scale phenomena.

Its key properties critical to our model are confirmed by the existing literature. The double-series converges absolutely for all finite $z,w\in \mathbb{R}$ [10], guaranteed by $(\mu -\nu )k+(\mu -\xi )m+\mu >0$ under $0<\xi <\nu <\mu \le 1$. For negative arguments in our model, it is bounded on $[0,T]$ [9], supporting integral kernel integrability.

For detailed treatments of the analytic properties, integral representations, and applications of multivariate Mittag–Leffler functions, we refer to [10,12,13].

Remark 3.

We define the multi-parameter Mittag–Leffler integral operator as

$${\mathbb{E}}_{\widehat{t}}^{\mu -\nu ,\mu -\xi ,\mu ,-e,-k}f(\widehat{t},x)={\int }_0^{\widehat{t}}{(\widehat{t}-s)}^{\mu -1}{E}_{\mu -\nu ,\mu -\xi ,\mu }\left(-e{(\widehat{t}-s)}^{\mu -\nu },-k{(\widehat{t}-s)}^{\mu -\xi }\right)f(s,x)ds,x\in \Omega$$

(4)

This operator plays a crucial role in constructing solutions to fractional differential equations with multiple damping terms. Recent advances in this operator-theoretic approach are detailed in [14,15]. It is well-defined, as ${(\widehat{t}-s)}^{\mu -1}$ is locally integrable, the Mittag–Leffler function is bounded and $f\in L_{loc}^1$ together ensuring the integrability.

3. Riemann–Liouville Fractional-Order ($\mathit{\mu },\mathit{\nu },\mathit{\xi },\mathit{e},\mathit{k}$) Resolvent

The theoretical analysis of Riemann–Liouville fractional evolution equations relies heavily on resolvent family theory, which serves as a bridge between abstract operator theory and the practical solution of fractional differential equations. Building upon the preliminary framework and the concept of strong solutions established earlier, this section extends the construction to the three-parameter damped case. We introduce the corresponding Riemann–Liouville fractional $(\mu ,\nu ,\xi ,e,k)$-resolvent operator and systematically analyze its properties.

Definition 5.

Let $0<\xi <\nu <\mu \le 1$ and let ${\left\{S\left(\widehat{t}\right)\right\}}_{\widehat{t}>0}$ be a family of linear operators acting on a Banach space $X=L^2(\Omega )$. This operator family is defined as the Riemann–Liouville fractional $(\mu ,\nu ,\xi ,e,k)$-resolvent family if it satisfies the following three core conditions:

• $\left(a\right)$ For every element $x\in \Omega$, the mapping $\widehat{t}\mapsto S\left(\widehat{t}\right)x_0\left(x\right)$ is strongly continuous on the interval $(0,\infty )$. Additionally, it satisfies the initial limit constraint characterizing the operator family’s behavior near $\widehat{t}=0$:

  $$\underset{\widehat{t}\to 0^{+}}{lim}\Gamma \left(\mu \right){\widehat{t}}^{1-\mu }S\left(\widehat{t}\right)x_0\left(x\right)=x_0\left(x\right),$$

  (5)
• $\left(b\right)$ The operators commute: $S\left(\widehat{t}\right)S\left(s\right)=S\left(s\right)S\left(\widehat{t}\right)$ for any pair of positive real numbers $\widehat{t}$ and s;
• $\left(c\right)$ For all $\widehat{t},s>0$, the following resolvent identity is derived as follows:

  $$\begin{array}{c}\hfill S\left(s\right){\mathbb{E}}_{\widehat{t}}^{\mu -\nu ,\mu -\xi ,\mu ,-e,-k}S\left(\widehat{t}\right)x_0\left(x\right)-{\mathbb{E}}_s^{\mu -\nu ,\mu -\xi ,\mu ,-e,-k}S\left(s\right)S\left(\widehat{t}\right)x_0\left(x\right)\\ \hfill =s^{\mu -1}{E}_{\mu -\nu ,\mu -\xi ,\mu }(-e{s}^{\mu -\nu },-k{s}^{\mu -\xi }){\mathbb{E}}_{\widehat{t}}^{\mu -\nu ,\mu -\xi ,\mu ,-e,-k}S\left(\widehat{t}\right)x_0\left(x\right)\\ \hfill -{\widehat{t}}^{\mu -1}{E}_{\mu -\nu ,\mu -\xi ,\mu }(-e{s}^{\mu -\nu },-k{s}^{\mu -\xi }){\mathbb{E}}_s^{\mu -\nu ,\mu -\xi ,\mu ,-e,-k}S\left(s\right)x_0\left(x\right).\end{array}$$

  (6)

  where ${\mathbb{E}}_{\widehat{t}}^{\mu -\nu ,\mu -\xi ,\mu ,-e,-k}f(\widehat{t},x)$ = ${\int }_0^{\widehat{t}}{(\widehat{t}-s)}^{\mu -1}{E}_{\mu -\nu ,\mu -\xi ,\mu }(-e{(\widehat{t}-s)}^{\mu -\nu },-k{(\widehat{t}-s)}^{\mu -\xi })f(s,x)ds$ is the multi-parameter Mittag–Leffler integral operator, and $E_{a,b,c}(z,w)={\sum }_{k=0}^{\infty }{\sum }_{m=0}^{\infty }{\displaystyle \frac{z^k{w}^m}{\Gamma (ak+bm+c)k!m!}}$ is the trivariate generalized Mittag–Leffler function.

This definition represents an extension of the $(\alpha ,\beta ,c)$-resolvent family proposed by Mei and Peng [1]. Compared with the original two-parameter resolvent family, this definition retains core properties such as strong continuity and commutativity by introducing the three-parameter Mittag–Leffler function, ensuring theoretical compatibility while adding the characterization of the lower-order damping effect corresponding to ξ.

Remark 4.

The integrals in Equation (6) are interpreted as strong Bochner integrals, a standard approach in Banach space theory that guarantees the well-definedness of integral operations for vector-valued functions. The composite integral term

$${\mathbb{E}}_{\widehat{t}}^{\mu -\nu ,\mu -\xi ,\mu ,-e,-k}S\left(\widehat{t}\right)x_0\left(x\right)$$

is the core of the three-parameter framework: its kernel is the trivariate generalized Mittag–Leffler function

$$E_{\mu -\nu ,\mu -\xi ,\mu }(-e{(\widehat{t}-s)}^{\mu -\nu },-k{(\widehat{t}-s)}^{\mu -\xi })$$

This kernel inherently encodes multi-scale damping effects through the parameters $e,k$, and fractional exponents $\mu -\nu$ and $\mu -\xi$. This structural design effectively bridges the gap between multi-parameter operator theory and fractional calculus applications. The Bochner integral is well-defined, as the integrand is strongly measurable and locally bounded, which satisfies the criterion in [8].

Remark 5.

Definition 5 is a natural generalization of the undamped $(\mu ,\nu ,\xi )$-resolvent family $(e=0,k=0)$ proposed in [1]. Notably, $S\left(\widehat{t}\right)$ is not defined at $\widehat{t}=0$, which arises from the singularity of the kernel component

$$\widehat{t}\mapsto {\widehat{t}}^{\mu -1}{E}_{\mu -\nu ,\mu -\xi ,\mu }(-e{\widehat{t}}^{\mu -\nu },-k{\widehat{t}}^{\mu -\xi })$$

at $\widehat{t}=0$. This singularity is intrinsic to Riemann–Liouville-type fractional operator families and aligns with the expected regularity of solutions of fractional differential equations.

Remark 6.

To simplify subsequent analysis, we introduce an auxiliary operator $\left(PS\right)(·)x_0\left(x\right)$ defined as

$$\left(PS\right)\left(\widehat{t}\right)x_0\left(x\right)=\left\{\begin{array}{cc}\Gamma \left(\mu \right){\widehat{t}}^{1-\mu }S\left(\widehat{t}\right)x_0\left(x\right),\hfill & \widehat{t}>0\hfill \\ x,\hfill & \widehat{t}=0\hfill \end{array}\right.$$

From Condition (a) of Definition 5, $\left(PS\right)\left(\widehat{t}\right)x_0\left(x\right)$ is continuous at $\widehat{t}=0$, and thus strongly continuous on $[0,\infty )$. Applying the Uniform Boundedness Principle to the strongly continuous operator-valued function $\left(PS\right)(·)x_0\left(x\right)$, we deduce that $\Gamma \left(\mu \right){\widehat{t}}^{1-\mu }S\left(\widehat{t}\right)x_0\left(x\right)$ is bounded on any finite interval $(0,b]$$(b>0)$.

Remark 7.

For any $x\in \Omega$, we have $S(·)x_0\left(x\right)\in L_{loc}^1([0,\infty )\times \Omega ;X)$. from condition $\left(a\right)$ of Definition 5 and the initial limit constraint $\underset{\widehat{t}\to 0^{+}}{lim}\Gamma \left(\mu \right){\widehat{t}}^{1-\mu }S\left(\widehat{t}\right)x_0\left(x\right)=x_0\left(x\right)$, it can be directly deduced that there exists a small positive constant $\delta >0$ such that for all $0<\widehat{t}<\delta$, the norm of $S\left(\widehat{t}\right)x_0\left(x\right)$ satisfies the inequality:

$$\parallel S\left(\widehat{t}\right)x_0\left(x\right)\parallel \le {\displaystyle \frac{2{\widehat{t}}^{\mu -1}}{\Gamma \left(\mu \right)}}\parallel x_0\left(x\right)\parallel$$

(7)

To verify the local integrability of $S(·)x_0\left(x\right)$, we take an arbitrary positive constant ${\tau }_0>0$ and decompose the integral ${\int }_0^{{\tau }_0}\parallel S\left(\widehat{t}\right)x_0\left(x\right)\parallel d\widehat{t}$ into two parts based on the interval $(0,\delta ]$ and $[\delta ,{\tau }_0]$. In the case where ${\tau }_0>\delta$, the integral is split into

$${\int }_0^{\delta }\parallel S\left(\widehat{t}\right)x_0\left(x\right)\parallel d\widehat{t}+{\int }_{\delta }^{{\tau }_0}\parallel S\left(\widehat{t}\right)x_0\left(x\right)\parallel d\widehat{t}$$

The first part converges as demonstrated; while the second part is bounded due to the strong continuity of $S\left(\widehat{t}\right)x_0\left(x\right)$ on $[\delta ,{\tau }_0]$. This implies that $S\left(\widehat{t}\right)x_0\left(x\right)$ is uniformly bounded on this interval, making the integral finite. Combining these two cases, we conclude that ${\int }_0^{{\tau }_0}\parallel S\left(\widehat{t}\right)x_0\left(x\right)\parallel d\widehat{t}<\infty$ for any ${\tau }_0>0$, which confirms $S(·)x_0\left(x\right)\in L_{loc}^1([0,\infty )\times \Omega ;X)$.

Definition 6.

Let ${\left\{S\left(\widehat{t}\right)\right\}}_{\widehat{t}>0}$ be a Riemann–Liouville fractional $(\mu ,\nu ,\xi ,e,k)$-resolvent family on $X=L^2(\Omega )$. The generator of $\left\{S\left(\widehat{t}\right)\right\}$ is a linear operator A: $D\left(A\right)\subset X\to X$, where its domain $D\left(A\right)$ and operator action are defined as follows:

Domain $D\left(A\right)$: Consists of initial state $x_0\left(x\right)\in X$ for which the limit exists in X

$$\underset{\widehat{t}\to 0^{+}}{lim}{\displaystyle \frac{{\widehat{t}}^{1-\mu }S\left(\widehat{t}\right)x_0\left(x\right)-E_{\mu -\nu ,\mu -\xi ,\mu }(-e{\widehat{t}}^{\mu -\nu },-k{\widehat{t}}^{\mu -\xi })x_0\left(x\right)}{{\widehat{t}}^{\mu }}}$$

For any $x_0\left(x\right)\in D\left(A\right)$ and all $x\in \Omega$,

$$A{x}_0\left(x\right)=\Gamma \left(2\mu \right)·\underset{\widehat{t}\to 0^{+}}{lim}{\displaystyle \frac{{\widehat{t}}^{1-\mu }S\left(\widehat{t}\right)x_0\left(x\right)-E_{\mu -\nu ,\mu -\xi ,\mu }(-e{\widehat{t}}^{\mu -\nu },-k{\widehat{t}}^{\mu -\xi })x_0\left(x\right)}{{\widehat{t}}^{\mu }}}$$

The generator A is a closed linear operator on X, that uniquely determines the resolvent family $\left\{S\left(\widehat{t}\right)\right\}$. In this case, A is called the generator of the resolvent family ${\left\{S\left(\widehat{t}\right)\right\}}_{\widehat{t}>0}$.

This definition is generalized from the generator of the two-parameter damped RLAFCP resolvent family [1] and the generator of the undamped fractional resolvent family [2].

Proposition 1.

Let ${\left\{S\left(\widehat{t}\right)\right\}}_{\widehat{t}>0}$ be a Riemann–Liouville fractional $(\mu ,\nu ,\xi ,e,k)$-resolvent family on $X=L^2(\Omega )$ with generator A. The following properties hold for all $\widehat{t}>0$ and $x\in \Omega$:

• $\left(a\right)$$S\left(\widehat{t}\right)D\left(A\right)\subset D\left(A\right)$ and $AS\left(\widehat{t}\right)x_0\left(x\right)=S\left(\widehat{t}\right)A{x}_0\left(x\right)$ for each $x_0\left(x\right)\in D\left(A\right)$.
• $\left(b\right)$ For each $x_0\left(x\right)\in X$ and $\widehat{t}>0$, recent advances in resolvent theory [16].

  $$S\left(\widehat{t}\right)x_0\left(x\right)={\widehat{t}}^{\mu -1}{E}_{\mu -\nu ,\mu -\xi ,\mu }(-e{\widehat{t}}^{\mu -\nu },-k{\widehat{t}}^{\mu -\xi })x_0\left(x\right)+A{\mathbb{E}}_{\widehat{t}}^{\mu -\nu ,\mu -\xi ,\mu ,-e,-k}S\left(\widehat{t}\right)x_0\left(x\right).$$

  (8)
• $\left(c\right)$ For any $x_0\left(x\right)\in D\left(A\right)$ and $\widehat{t}>0$, the decomposition form of $S\left(\widehat{t}\right)$ can be further adjusted by exchanging the order of A and the integral operator:

  $$S\left(\widehat{t}\right)x_0\left(x\right)={\widehat{t}}^{\mu -1}{E}_{\mu -\nu ,\mu -\xi ,\mu }(-e{\widehat{t}}^{\mu -\nu },-k{\widehat{t}}^{\mu -\xi })x_0\left(x\right)+{\mathbb{E}}_{\widehat{t}}^{\mu -\nu ,\mu -\xi ,\mu ,-e,-k}S\left(\widehat{t}\right)A{x}_0\left(x\right).$$

  (9)
• $\left(d\right)$ The generator A is a closed linear operator on $X=L^2(\Omega )$.

Proof. 

$\left(a\right)$ Let $x_0\left(x\right)\in D\left(A\right)$, we utilize the commutativity property from Definition 5$\left(b\right)$ along with strong continuity of $S(·)x_0\left(x\right)$ on $(0,\infty )$. The key step is to interchange the limit operation with the operator action of $S\left(\widehat{t}\right)$, For the interchange of the limit and the operator $S\left(\widehat{t}\right)$, we rely on the property that bounded linear operators commute with strongly convergent sequences ([3], Section 2.3)

$$\begin{array}{cc}& \Gamma \left(2\mu \right)\underset{s\to 0^{+}}{lim}{\displaystyle \frac{s^{1-\mu }S\left(s\right)S\left(\widehat{t}\right)x_0\left(x\right)-E_{\mu -\nu ,\mu -\xi ,\mu }(-e{s}^{\mu -\nu },-k{s}^{\mu -\xi })S\left(\widehat{t}\right)x_0\left(x\right)}{{\widehat{t}}^{\mu }}}\hfill \\ & =S\left(\widehat{t}\right)\Gamma \left(2\mu \right)\underset{s\to 0^{+}}{lim}{\displaystyle \frac{s^{1-\mu }S\left(s\right)x_0\left(x\right)-E_{\mu -\nu ,\mu -\xi ,\mu }(-e{s}^{\mu -\nu },-k{s}^{\mu -\xi })x_0\left(x\right)}{s^{\mu }}}\hfill \\ & =S\left(\widehat{t}\right)A{x}_0\left(x\right).\hfill \end{array}$$

The existence of the left-hand limit implies that $S\left(\widehat{t}\right)x_0\left(x\right)$ satisfies the domain condition of A, and the equality $AS\left(\widehat{t}\right)x_0\left(x\right)=S\left(\widehat{t}\right)A{x}_0\left(x\right)$ follows directly from the above formula. Thus, property $\left(a\right)$ holds.

• $\left(b\right)$ For any $x_0\left(x\right)\in X$, we have

  $$\begin{array}{cccc}& ∥\Gamma \left(2\mu \right){\displaystyle \frac{\mathbb{E}{s}^{\mu -\nu ,\mu -\xi ,\mu ,-e,-k}S\left(s\right)x_0\left(x\right)}{s^{2\mu -1}}}-x_0\left(x\right)∥\hfill & & \\ & =∥\Gamma \left(2\mu \right){\int }_0^s{(s-\sigma )}^{\mu -1}{s}^{1-2\mu }{E}_{\mu -\nu ,\mu -\xi ,\mu }\left(-e{(s-\sigma )}^{\mu -\nu },-k{(s-\sigma )}^{\mu -\xi }\right)S\left(\sigma \right)x_0\left(x\right)d\sigma -x_0\left(x\right)∥\hfill & & \\ & =∥\Gamma \left(2\mu \right){\int }_0^1{(1-\sigma )}^{\mu -1}{s}^{1-\mu }{E}_{\mu -\nu ,\mu -\xi ,\mu }\left(-e{(s-s\sigma )}^{\mu -\nu },-k{(s-s\sigma )}^{\mu -\xi }\right)S\left(s\sigma \right)x_0\left(x\right)d\sigma -x_0\left(x\right)∥\hfill & & \\ & =∥{\displaystyle \frac{\Gamma \left(2\mu \right)}{\Gamma \left(\mu \right)}}{\int }_0^1{(1-\sigma )}^{\mu -1}{\sigma }^{\mu -1}{E}_{\mu -\nu ,\mu -\xi ,\mu }\left(-e{(s-s\sigma )}^{\mu -\nu },-k{(s-s\sigma )}^{\mu -\xi }\right)\Gamma \left(\mu \right){\left(s\sigma \right)}^{1-\mu }S\left(s\sigma \right)x_0\left(x\right)d\sigma -x_0\left(x\right)∥\hfill & & \\ & =∥{\displaystyle \frac{\Gamma \left(2\mu \right)}{\Gamma \left(\mu \right)\Gamma \left(\mu \right)}}{\int }_0^1{(1-\sigma )}^{\mu -1}{\sigma }^{\mu -1}\Gamma \left(\mu \right)E_{\mu -\nu ,\mu -\xi ,\mu }\left(-e{(s-s\sigma )}^{\mu -\nu },-k{(s-s\sigma )}^{\mu -\xi }\right){\left(s\sigma \right)}^{1-\mu }\Gamma \left(\mu \right)S\left(s\sigma \right)x_0\left(x\right)d\sigma \hfill & & \\ & -{\displaystyle \frac{\Gamma \left(2\mu \right)}{\Gamma \left(\mu \right)\Gamma \left(\mu \right)}}{\int }_0^1{(1-\sigma )}^{\mu -1}{\sigma }^{\mu -1}{x}_0\left(x\right)d\sigma ∥\hfill & & \\ & \le {\displaystyle \frac{\Gamma \left(2\mu \right)}{\Gamma \left(\mu \right)\Gamma \left(\mu \right)}}{\int }_0^1{(1-\sigma )}^{\mu -1}{\sigma }^{\mu -1}d\sigma \hfill & & \\ & \times \underset{\sigma \in (0,1]}{sup}∥\Gamma \left(\mu \right)E_{\mu -\nu ,\mu -\xi ,\mu }\left(-e{(s-s\sigma )}^{\mu -\nu },-k{(s-s\sigma )}^{\mu -\xi }\right)\Gamma \left(\mu \right){\left(s\sigma \right)}^{1-\mu }S\left(s\sigma \right)x_0\left(x\right)-x_0\left(x\right)∥\hfill & & \\ & =\underset{\sigma \in (0,1]}{sup}∥\Gamma \left(\mu \right)E_{\mu -\nu ,\mu -\xi ,\mu }\left(-e{(s-s\sigma )}^{\mu -\nu },-k{(s-s\sigma )}^{\mu -\xi }\right)\Gamma \left(\mu \right){\left(s\sigma \right)}^{1-\mu }S\left(s\sigma \right)x_0\left(x\right)-x_0\left(x\right)∥\hfill & \end{array}$$

  (10)

  where $s\mapsto E_{\mu -\nu ,\mu -\xi ,\mu }\left(-e{s}^{\mu -\nu }-k{s}^{\mu -\xi }\right)$ is uniformly continuous on $[0,1]$ and

  $$E_{\mu -\nu ,\mu -\xi ,\mu }\left(-e·0^{\mu -\nu }-k·0^{\mu -\xi }\right)={\displaystyle \frac{1}{\Gamma \left(\mu \right)}}$$

  The combination of $\left(9\right)$, $\left(10\right)$, $\left(a\right)$ of Definition 5 implies that

  $$\underset{s\to 0^{+}}{lim}\Gamma \left(2\mu \right){\displaystyle \frac{{\mathbb{E}}_s^{\mu -\nu ,\mu -\xi ,\mu ,-e,-k}S\left(s\right)x_0\left(x\right)}{s^{2\mu -1}}}=1$$

  (11)

  For the interchange in the Bochner integral, we observe that the integrand is integrable, converges pointwise, and has a control function, as justified in [2]. By Definition 5 and $\left(11\right)$, we obtain the following:

  $$\begin{array}{cc}& A{\mathbb{E}}_{\widehat{t}}^{\mu -\nu ,\mu -\xi ,\mu ,-e,-k}S\left(\widehat{t}\right)x_0\left(x\right)\hfill \\ & =\Gamma \left(2\mu \right)\underset{s\to 0^{+}}{lim}{\displaystyle \frac{s^{1-\mu }S\left(s\right){\mathbb{E}}_{\widehat{t}}^{\mu -\nu ,\mu -\xi ,\mu ,-e,-k}S\left(\widehat{t}\right)x_0\left(x\right)-E_{\mu -\nu ,\mu -\xi ,\mu }\left(-e{s}^{\mu -\nu },-k{s}^{\mu -\xi }\right){\mathbb{E}}_{\widehat{t}}^{\mu -\nu ,\mu -\xi ,\mu ,-e,-k}S\left(\widehat{t}\right)x_0\left(x\right)}{s^{\mu }}}\hfill \\ & =\Gamma \left(2\mu \right)\underset{s\to 0^{+}}{lim}{\displaystyle \frac{{\mathbb{E}}_s^{\mu -\nu ,\mu -\xi ,\mu ,-e,-k}S\left(s\right)\left(S\left(\widehat{t}\right)x_0\left(x\right)-{\widehat{t}}^{\mu -1}{E}_{\mu -\nu ,\mu -\xi ,\mu }\left(-e{\widehat{t}}^{\mu -\nu },-k{\widehat{t}}^{\mu -\xi }\right)x_0\left(x\right)\right)}{s^{2\mu -1}}}\hfill \\ & =S\left(\widehat{t}\right)x_0\left(x\right)-{\widehat{t}}^{\mu -1}{E}_{\mu -\nu ,\mu -\xi ,\mu }\left(-e{\widehat{t}}^{\mu -\nu },-k{\widehat{t}}^{\mu -\xi }\right)x_0\left(x\right)\hfill \end{array}$$

  therefore, it can be inferred that $\left(b\right)$ holds.
• $\left(c\right)$ Let $x\in D\left(A\right).$ According to Definition 6, the limit

  $$\underset{\widehat{t}\to 0^{+}}{lim}{\displaystyle \frac{{\widehat{t}}^{1-\mu }S\left(\widehat{t}\right)x_0\left(x\right)-E_{\mu -\nu ,\mu -\xi ,\mu }\left(-e{\widehat{t}}^{\mu -\nu },-k{\widehat{t}}^{\mu -\xi }\right)x_0\left(x\right)}{{\widehat{t}}^{\mu }}}$$

  exists, which implies that the function

  $$g\left(s\right)={\displaystyle \frac{{\widehat{t}}^{1-\mu }S\left(\widehat{t}\right)x_0\left(x\right)-E_{\mu -\nu ,\mu -\xi ,\mu }\left(-e{\widehat{t}}^{\mu -\nu },-k{\widehat{t}}^{\mu -\xi }\right)x_0\left(x\right)}{{\widehat{t}}^{\mu }}}$$

  is bounded for all sufficiently small values of $s>0$. To justify the interchange of the limit and the integral, we note that the integrand is Bochner measurable, admits an $L^1$ control function, and converges strongly pointwise, which follows the dominated convergence theorem [5] Chapter 3. For any $\widehat{t}>0$, we apply the dominated convergence theorem to justify the interchange of limit and integral operations

  $$\begin{array}{cc}& S\left(\widehat{t}\right)x_0\left(x\right)-{\widehat{t}}^{\mu -1}{E}_{\mu -\nu ,\mu -\xi ,\mu }\left(-e{\widehat{t}}^{\mu -\nu },-k{\widehat{t}}^{\mu -\xi }\right)x_0\left(x\right)\hfill \\ & =A{\mathbb{E}}_{\widehat{t}}^{\mu -\nu ,\mu -\xi ,\mu ,-e,-k}S\left(\widehat{t}\right)x_0\left(x\right)\hfill \\ & =\Gamma \left(2\mu \right)\underset{s\to 0^{+}}{lim}{\displaystyle \frac{s^{1-\mu }S\left(s\right)-E_{\mu -\nu ,\mu -\xi ,\mu }\left(-e{s}^{\mu -\nu },-k{s}^{\mu -\xi }\right)}{s^{\mu }}}\hfill \\ & ·{\int }_0^{\widehat{t}}{(\widehat{t}-\sigma )}^{\mu -1}{E}_{\mu -\nu ,\mu -\xi ,\mu }\left(-e{(\widehat{t}-\sigma )}^{\mu -\nu },-k{(\widehat{t}-\sigma )}^{\mu -\xi }\right)S\left(\sigma \right)x_0\left(x\right)d\sigma \hfill \\ & =\Gamma \left(2\mu \right)\underset{s\to 0^{+}}{lim}{\int }_0^{\widehat{t}}{(\widehat{t}-\sigma )}^{\mu -1}{E}_{\mu -\nu ,\mu -\xi ,\mu }\left(-e{(\widehat{t}-\sigma )}^{\mu -\nu },-k{(\widehat{t}-\sigma )}^{\mu -\xi }\right)\hfill \\ & ·S\left(\sigma \right){\displaystyle \frac{s^{1-\mu }S\left(s\right)-E_{\mu -\nu ,\mu -\xi ,\mu }\left(-e{s}^{\mu -\nu },-k{s}^{\mu -\xi }\right)}{s^{\mu }}}{x}_0\left(x\right)d\sigma \hfill \\ & ={\mathbb{E}}_{\widehat{t}}^{\mu -\nu ,\mu -\xi ,\mu ,-e,-k}S\left(\widehat{t}\right)A{x}_0\left(x\right)\hfill \end{array}$$

  which proves (9).
• $\left(d\right)$ To prove A is closed: Let ${\left\{x_{0,n}\left(x\right)\right\}}_{n=1}^{\infty }\subset D\left(A\right)$ satisfy $x_{0,n}\left(x\right)\to x_0\left(x\right)$ and $A{x}_{0,n}\left(x\right)\to y\left(x\right)$ as $n\to \infty$. By property $\left(c\right)$, for any $\widehat{t}>0$:

  $$\begin{array}{cc}& S\left(\widehat{t}\right)x_0\left(x\right)-{\widehat{t}}^{\mu -1}{E}_{\mu -\nu ,\mu -\xi ,\mu }\left(-e{\widehat{t}}^{\mu -\nu },-k{\widehat{t}}^{\mu -\xi }\right)x_0\left(x\right)\hfill \\ & =\underset{n\to \infty }{lim}\left(S\left(\widehat{t}\right)x_{0,n}\left(x\right)-{\widehat{t}}^{\mu -1}{E}_{\mu -\nu ,\mu -\xi ,\mu }\left(-e{\widehat{t}}^{\mu -\nu },-k{\widehat{t}}^{\mu -\xi }\right)x_{0,n}\left(x\right)\right)\hfill \\ & =\underset{n\to \infty }{lim}{\int }_0^{\widehat{t}}{(\widehat{t}-\sigma )}^{\mu -1}{E}_{\mu -\nu ,\mu -\xi ,\mu }\left(-e{(\widehat{t}-\sigma )}^{\mu -\nu },-k{(\widehat{t}-\sigma )}^{\mu -\xi }\right)S\left(\sigma \right)A{x}_{0,n}\left(x\right)d\tau \hfill \\ & ={\int }_0^{\widehat{t}}{(\widehat{t}-\sigma )}^{\mu -1}{E}_{\mu -\nu ,\mu -\xi ,\mu }\left(-e{(\widehat{t}-\sigma )}^{\mu -\nu },-k{(\widehat{t}-\sigma )}^{\mu -\xi }\right)S\left(\sigma \right)Ay\left(x\right)d\tau \hfill \\ & ={\mathbb{E}}_{\widehat{t}}^{\mu -\nu ,\mu -\xi ,\mu ,-e,-k}S\left(\widehat{t}\right)y\left(x\right).\hfill \end{array}$$

  Using $\left(11\right)$ have

  $$\begin{array}{cc}\hfill A{x}_0\left(x\right)& =\Gamma \left(2\mu \right)\underset{\widehat{t}\to 0^{+}}{lim}{\displaystyle \frac{{\widehat{t}}^{1-\mu }S\left(\widehat{t}\right)x_0\left(x\right)-E_{\mu -\nu ,\mu -\xi ,\mu }\left(-e{\widehat{t}}^{\mu -\nu },-k{\widehat{t}}^{\mu -\xi }\right)x_0\left(x\right)}{{\widehat{t}}^{\mu }}}\hfill \\ & =\Gamma \left(2\mu \right)\underset{\widehat{t}\to 0^{+}}{lim}{\displaystyle \frac{S\left(\widehat{t}\right)x_0\left(x\right)-{\widehat{t}}^{\mu -1}{E}_{\mu -\nu ,\mu -\xi ,\mu }\left(-e{\widehat{t}}^{\mu -\nu },-k{\widehat{t}}^{\mu -\xi }\right)x_0\left(x\right)}{{\widehat{t}}^{2\mu -1}}}\hfill \\ & =\Gamma \left(2\mu \right)\underset{\widehat{t}\to 0^{+}}{lim}{\displaystyle \frac{{\mathbb{E}}_{\widehat{t}}^{\mu -\nu ,\mu -\xi ,\mu ,-e,-k}S\left(\widehat{t}\right)y\left(x\right)}{{\widehat{t}}^{2\mu -1}}}\hfill \\ & =y\left(x\right)\hfill \end{array}$$

  By the definition of the generator A, this implies that $x_0\left(x\right)\in D\left(A\right)$ and $A{x}_0\left(x\right)=y\left(x\right)$. This verifies that A satisfies the definition of a closed linear operator on X. □

Theorem 1.

Let ${\left\{S\left(\widehat{t}\right)\right\}}_{\widehat{t}>0}$ and ${\left\{U\left(\widehat{t}\right)\right\}}_{\widehat{t}>0}$ be Riemann–Liouville fractional $(\mu ,\nu ,\xi ,e,k)$ resolvent families defined on a Banach space $X=L^2(\Omega )$, with generators A and B, respectively. If the generators satisfy A = B, then the resolvent families are identical for all positive times: $S\left(\widehat{t}\right)=U\left(\widehat{t}\right)$ for every $\widehat{t}>0$.

Proof. 

For any initial state $x_0\left(x\right)\in D\left(A\right)$ and $x\in \Omega$, substituting the decomposition formula $\left(9\right)$ of Proposition 1 into the convolution operation of the resolvent families, we obtain the following:

$$\begin{array}{cc}& {\widehat{t}}^{\mu -1}{E}_{\mu -\nu ,\mu -\xi ,\mu }\left(-e{\widehat{t}}^{\mu -\nu },-k{\widehat{t}}^{\mu -\xi }\right)\ast U\left(\widehat{t}\right)x_0\left(x\right)\hfill \\ & =\left(S\left(\widehat{t}\right)-{\mathbb{E}}_{\widehat{t}}^{\mu -\nu ,\mu -\xi ,\mu ,-e,-k}AS\left(\widehat{t}\right)\right)\ast U\left(\widehat{t}\right)x_0\left(x\right)\hfill \\ & =S\left(\widehat{t}\right)\ast U\left(\widehat{t}\right)x_0\left(x\right)-{\widehat{t}}^{\mu -1}{E}_{\mu -\nu ,\mu -\xi ,\mu }\left(-e{\widehat{t}}^{\mu -\nu },-k{\widehat{t}}^{\mu -\xi }\right)\ast AS\left(\widehat{t}\right)\ast U\left(\widehat{t}\right)x_0\left(x\right)\hfill \\ & =S\left(\widehat{t}\right)\ast U\left(\widehat{t}\right)x_0\left(x\right)-{\widehat{t}}^{\mu -1}{E}_{\mu -\nu ,\mu -\xi ,\mu }\left(-e{\widehat{t}}^{\mu -\nu },-k{\widehat{t}}^{\mu -\xi }\right)\ast S\left(\widehat{t}\right)\ast AU\left(\widehat{t}\right)x_0\left(x\right)\hfill \\ & =S\left(\widehat{t}\right)\ast \left(U\left(\widehat{t}\right)x_0\left(x\right)-{\widehat{t}}^{\mu -1}{E}_{\mu -\nu ,\mu -\xi ,\mu }\left(-e{\widehat{t}}^{\mu -\nu },-k{\widehat{t}}^{\mu -\xi }\right)\ast AU\left(\widehat{t}\right)x_0\left(x\right)\right)\hfill \\ & =S\left(\widehat{t}\right)\ast {\widehat{t}}^{\mu -1}{E}_{\mu -\nu ,\mu -\xi ,\mu }\left(-e{\widehat{t}}^{\mu -\nu },-k{\widehat{t}}^{\mu -\xi }\right)x_0\left(x\right)\hfill \\ & ={\widehat{t}}^{\mu -1}{E}_{\mu -\nu ,\mu -\xi ,\mu }\left(-e{\widehat{t}}^{\mu -\nu },-k{\widehat{t}}^{\mu -\xi }\right)\ast S\left(\widehat{t}\right)x_0\left(x\right)\hfill \end{array}$$

Next, we invoke Titchmarsh’s convolution theorem: if the convolution of continuous functions $f\ast g=f\ast h$ holds, and the function g does not vanish on its domain, then g and h are equal almost everywhere. The function $f\left(\widehat{t}\right)={\widehat{t}}^{\mu -1}{E}_{\mu -\nu ,\mu -\xi ,\mu }\left(-e{\widehat{t}}^{\mu -\nu },-k{\widehat{t}}^{\mu -\xi }\right)$ is continuous and non-vanishing for $\widehat{t}>0$, so the convolution identity implies: $S\left(\widehat{t}\right)x_0\left(x\right)=U\left(\widehat{t}\right)x_0\left(x\right)$ for all $\widehat{t}>0$, $x\in \Omega$ and $x_0\left(x\right)\in D\left(A\right)$, regarding the interchange of the limit and the operators $S\left(\widehat{t}\right)$ and $U\left(\widehat{t}\right)$, we use the fact that uniformly bounded operators can be extended from dense subsets via strong convergence [8] Lemma 2.3.

Finally, since $D\left(A\right)$ is a dense subset of the Banach space $X=L^2(\Omega )$, for any initial state $x_0\left(x\right)\in X$ and $x\in \Omega$, there always exists a sequence ${\left\{x_{0,n}\left(x\right)\right\}}_{n=1}^{\infty }\subset D\left(A\right)$ such that $x_{0,n}\left(x\right)\to x_0\left(x\right)$ as $n\to \infty$. By leveraging the strong continuity and boundedness of $S\left(\widehat{t}\right)$ and $U\left(\widehat{t}\right)$, we get: $S\left(\widehat{t}\right)x_0\left(x\right)=\underset{n\to \infty }{lim}S\left(\widehat{t}\right)x_{0,n}\left(x\right)=\underset{n\to \infty }{lim}U\left(\widehat{t}\right)x_{0,n}\left(x\right)=U\left(\widehat{t}\right)x_0\left(x\right)$.

In summary, the resolvent families $S\left(\widehat{t}\right)$ and $U\left(\widehat{t}\right)$ are identical for all $\widehat{t}>0$ and $x\in \Omega$, which completes the proof of Theorem 1. □

Remark 8.

Theorem 1 establishes that every closed and densely defined linear operator A generates at most one Riemann–Liouville fractional $(\mu ,\nu ,\xi ,e,k)$-resolvent family, consistent with the results in [8].

Theorem 2.

Let $A:D\left(A\right)\subset X\to X$ be a closed and densely defined linear operator on a Banach space $X=L^2(\Omega )$. Then A generates an exponentially bounded Riemann–Liouville fractional $(\mu ,\nu ,\xi ,e,k)$-resolvent family ${\left\{S\left(\widehat{t}\right)\right\}}_{\widehat{t}>0}$ if and only if the following four conditions hold simultaneously:

• $\left(2.1\right)$ For every initial state $x_0\left(x\right)\in X$, the mapping $\widehat{t}\mapsto S\left(\widehat{t}\right)x_0\left(x\right)$ is continuous on $(0,\infty )$, and the initial limit condition is satisfied for all $x\in \Omega$:

  $$\underset{\widehat{t}\to 0^{+}}{lim}\Gamma \left(\mu \right){\widehat{t}}^{1-\mu }S\left(\widehat{t}\right)x_0\left(x\right)=x_0\left(x\right)$$

  (12)
• $\left(2.2\right)$ The resolvent family is commutative for all positive times: $S\left(\widehat{t}\right)S\left(s\right)=S\left(s\right)S\left(\widehat{t}\right)$, for any $\widehat{t},s>0$ and $x\in \Omega$;.
• $\left(2.3\right)$ There exist constants $M\ge 1$ and $\omega \ge 0$ such that the operator norm satisfies $\parallel S\left(\widehat{t}\right)\parallel \le M{e}^{\omega \widehat{t}}$ for all $\widehat{t}>0$.
• $\left(2.4\right)$ For every $\lambda >\omega$, $x\in \Omega$ and all $x_0\left(x\right)\in X$, the resolvent operator of A admits the following integral representation via the Laplace transform of $S\left(\widehat{t}\right)$:

  $$R({\lambda }^{\mu }+e{\lambda }^{\nu }+k{\lambda }^{\xi },A)x_0\left(x\right)={\int }_0^{\infty }{e}^{-\lambda \widehat{t}}S\left(\widehat{t}\right)x_0\left(x\right)d\widehat{t}.$$

  (13)

Proof. 

(Necessity). Assume ${\left\{S\left(\widehat{t}\right)\right\}}_{\widehat{t}>0}$ is an exponentially bounded $(\mu ,\nu ,\xi ,e,k)$-resolvent family generated by A, satisfying $\parallel S\left(\widehat{t}\right)\parallel \le M{e}^{\omega \widehat{t}}$ for all $\widehat{t}>0$.

• $\left(2.1\right)$ The continuity of $S(·)x_0\left(x\right)$ at ${\widehat{t}}_0>0$ is established using the strong continuity of the resolvent family and the semigroup property:

  $$S\left(\widehat{t}\right)=S(\widehat{t}-{\widehat{t}}_0)S\left({\widehat{t}}_0\right)$$

  Combining this with the initial limit $\underset{s\to 0^{+}}{lim}S\left(s\right)x={\displaystyle \frac{{\widehat{t}}_0^{\mu -1}}{\Gamma \left(\mu \right)}}{x}_0\left(x\right)$, we estimate the norm difference:

  $$\parallel S\left(\widehat{t}\right)x_0\left(x\right)-S\left({\widehat{t}}_0\right)x_0\left(x\right)\parallel$$

  $$=\parallel S\left({\widehat{t}}_0\right)(S(\widehat{t}-{\widehat{t}}_0)x_0\left(x\right)-S\left(0^{+}\right)x_0\left(x\right))\parallel \le \parallel S\left({\widehat{t}}_0\right)\parallel ·\parallel S(\widehat{t}-{\widehat{t}}_0)x_0\left(x\right)-S\left(0^{+}\right)x_0\left(x\right)\parallel$$

  Thus, proving $S(·)x_0\left(x\right)$ is continuous at ${\widehat{t}}_0>0$.
• $\left(2.2\right)$ The commutation property follows directly from Definition 5$\left(b\right)$.
• $\left(2.3\right)$ By definition of the operator class $A\in C^{\mu ,\nu ,\xi }(M,\omega )$, standard classification for generators of exponentially bounded fractional resolvent families ensures the existence of universal constants $M\ge 1$ and $\omega \ge 0$ such that the norm bound $\parallel S\left(\widehat{t}\right)\parallel \le M{e}^{\omega \widehat{t}}$ holds for all $\widehat{t}>0$.
• $\left(2.4\right)$ apply the Laplace transform of both sides of equation, denote $\widehat{u}(\lambda ,x)=\mathcal{L}\left\{u(\widehat{t},x)\right\}$$\left(\lambda \right)$ [17] □

Laplace transform of the left-hand side

$$\begin{array}{cc}& \mathcal{L}\{D_{\widehat{t}}^{\mu }u+e{D}_{\widehat{t}}^{\nu }u+k{D}_{\widehat{t}}^{\xi }u\}\left(\lambda \right)\hfill \\ & ={\lambda }^{\mu }\widehat{u}\left(\lambda \right)-(G_{1-\mu }\ast u)\left(0\right)+e\left({\lambda }^{\nu }\widehat{u}\left(\lambda \right)-(G_{1-\nu }\ast u)\left(0\right)\right)+k({\lambda }^{\xi }\widehat{u}\left(\lambda \right)-(G_{1-\xi }\ast u)\left(0\right))\hfill \end{array}$$

Substituting the initial conditions $\underset{\widehat{t}\to 0^{+}}{lim}(G_{1-\mu }\ast u)(\widehat{t},x)=x_0\left(x\right)$, $\underset{\widehat{t}\to 0^{+}}{lim}(G_{1-\nu }\ast u)(\widehat{t},x)=0$, and $\underset{\widehat{t}\to 0^{+}}{lim}(G_{1-\xi }\ast u)(\widehat{t},x)=0$, we obtain

$$({\lambda }^{\mu }+e{\lambda }^{\nu }+k{\lambda }^{\xi })\widehat{u}\left(\lambda \right)-x_0\left(x\right)$$

Laplace transform of the right-hand side, we obtain

$$\mathcal{L}\left\{Au(\widehat{t},x)\right\}\left(\lambda \right)=A\widehat{u}(\lambda ,x)$$

The two sides of the equation are equal:

$$({\lambda }^{\mu }+e{\lambda }^{\nu }+k{\lambda }^{\xi })\widehat{u}(\lambda ,x)-x_0\left(x\right)=A\widehat{u}(\lambda ,x)$$

Rearranging and simplifying

$$\widehat{u}(\lambda ,x)={({\lambda }^{\mu }+e{\lambda }^{\nu }+k{\lambda }^{\xi }-A)}^{-1}{x}_0\left(x\right)=R\left({\lambda }^{\mu }+e{\lambda }^{\nu }+k{\lambda }^{\xi },A\right)x_0\left(x\right)$$

the strong solution to the modified problem is $u(\widehat{t},x)=S\left(\widehat{t}\right)x_0\left(x\right)$. Thus,

$$\widehat{u}(\lambda ,x)=\mathcal{L}\left\{S\left(\widehat{t}\right)x_0\left(x\right)\right\}\left(\lambda \right)={\int }_0^{\infty }{e}^{-\lambda \widehat{t}}S\left(\widehat{t}\right)x_0\left(x\right)d\widehat{t}$$

Therefore, for $x_0\left(x\right)\in D\left(A\right)$ and $x\in \Omega$, we have

$$R\left({\lambda }^{\mu }+e{\lambda }^{\nu }+k{\lambda }^{\xi },A\right)x_0\left(x\right)={\int }_0^{\infty }{e}^{-\lambda \widehat{t}}S\left(\widehat{t}\right)x_0\left(x\right)d\widehat{t}$$

Furthermore, since $D\left(A\right)$ is dense in X, for any $x_0\left(x\right)\in X$, take ${\left\{x_{0,n}\left(x\right)\right\}}_{n=1}^{\infty }\subset D\left(A\right)$ such that

$$\underset{n\to \infty }{lim}{x}_{0,n}\left(x\right)=x_0\left(x\right)$$

From the resolvent operator $R(\zeta ,A)$ is strongly continuous on $\zeta \in \rho \left(A\right)$, so

$$\underset{n\to \infty }{lim}R(\zeta ,A)x_{0,n}\left(x\right)=R(\zeta ,A)x_0\left(x\right)$$

From $\parallel S\left(\widehat{t}\right)x_{0,n}\left(x\right)-S\left(\widehat{t}\right)x_0\left(x\right)\parallel \le M{e}^{\omega \widehat{t}}\parallel x_{0,n}\left(x\right)-x_0\left(x\right)\parallel$, combined with the Dominated Convergence Theorem, we have the following:

$$\underset{n\to \infty }{lim}{\int }_0^{\infty }{e}^{-\lambda \widehat{t}}S\left(\widehat{t}\right)x_{0,n}\left(x\right)d\widehat{t}={\int }_0^{\infty }{e}^{-\lambda \widehat{t}}S\left(\widehat{t}\right)x_0\left(x\right)d\widehat{t}$$

Thus, the equation holds for all $x_0\left(x\right)\in X$ and $x\in \Omega$. Since $f\left(\lambda \right)$ is strictly increasing on $\lambda >0$,

$$f\left(\lambda \right)={\lambda }^{\mu }+e{\lambda }^{\nu }+k{\lambda }^{\xi }$$

its derivative

$$f^{\prime }\left(\lambda \right)=\mu {\lambda }^{\mu -1}+e\nu {\lambda }^{\nu -1}+k\xi {\lambda }^{\xi -1}>0$$

and as $\lambda \to {\omega }^{+}$, $f\left(\lambda \right)\to f\left(\omega \right)$; as $\lambda \to \infty$, $f\left(\lambda \right)\to \infty$.

Therefore, by the Intermediate Value Theorem, for any $\zeta >f\left(\omega \right)$, there exists a unique $\lambda >\omega$ such that $\zeta =f\left(\lambda \right)$. Furthermore, when $\lambda >\omega$, $R(f\left(\lambda \right),A)={\int }_0^{\infty }{e}^{-\lambda \widehat{t}}T\left(\widehat{t}\right)d\widehat{t}$ is bounded, so $f\left(\lambda \right)\in \rho \left(A\right)$. Since $\zeta =f\left(\lambda \right)$, we have $\zeta \in \rho \left(A\right)$, i.e., $\left(f\right(\omega ),\infty )\subset \rho \left(A\right)$.

(Sufficiency)

• $\left(2.1\right)$ By Definition 5 $\left(a\right)$, $S(·)x_0\left(x\right)\in C((0,\infty );X)$ and $\underset{\widehat{t}\to 0^{+}}{lim}\Gamma \left(\mu \right){\widehat{t}}^{1-\mu }S\left(\widehat{t}\right)x_0\left(x\right)=x_0\left(x\right)$ for all $x_0\left(x\right)\in X$ and $x\in \Omega$.
• $\left(2.2\right)$ It is directly given by Definition 5 $\left(2\right)$ that $S\left(\widehat{t}\right)S\left(s\right)=S\left(s\right)S\left(\widehat{t}\right)$, for $\widehat{t}$ and $s>0$.
• $(2.3,2.4)$ For all $\lambda >\omega$ and $\mu >\omega$, the resolvent operator satisfies the identity,

  $$R({\zeta }_1,A)-R({\zeta }_2,A)=({\zeta }_2-{\zeta }_1)R({\zeta }_1,A)R({\zeta }_2,A)$$

  where ${\zeta }_1$ = ${\lambda }^{\mu }+e{\lambda }^{\nu }+k{\lambda }^{\xi }$ and ${\zeta }_2$ = ${\mu }^{\mu }+e{\mu }^{\nu }+k{\mu }^{\xi }$.

Further, substituting $R({\zeta }_1,A)={\int }_0^{\infty }{e}^{-\lambda \widehat{t}}T\left(\widehat{t}\right)d\widehat{t}$ and $R({\zeta }_2,A)={\int }_0^{\infty }{e}^{-\mu s}T\left(s\right)ds$ into the resolvent identity

$${\int }_0^{\infty }{e}^{-\lambda \widehat{t}}S\left(\widehat{t}\right)d\widehat{t}-{\int }_0^{\infty }{e}^{-\mu s}S\left(s\right)ds=({\zeta }_2-{\zeta }_1){\int }_0^{\infty }{e}^{-\lambda \widehat{t}}S\left(\widehat{t}\right)d\widehat{t}{\int }_0^{\infty }{e}^{-\mu s}S\left(s\right)ds$$

By applying the Inverse Laplace Transform to both sides and utilizing the convolution property of the Laplace transform, we obtain the result that

$${\mathcal{L}}^{-1}\left\{R({\zeta }_1,A)R({\zeta }_2,A)\right\}\left(\widehat{t}\right)=(S\ast S)\left(\widehat{t}\right)={\int }_0^{\widehat{t}}S(\widehat{t}-\tau )S\left(\tau \right)d\tau .$$

Applying the inverse Laplace transform to the right-hand side

$${\mathcal{L}}^{-1}\left\{({\zeta }_2-{\zeta }_1)R({\zeta }_1,A)R({\zeta }_2,A)\right\}\left(\widehat{t}\right)=D_{\widehat{t}}^{\mu }(S\ast S)\left(\widehat{t}\right)+e{D}_{\widehat{t}}^{\nu }(S\ast S)\left(\widehat{t}\right)+k{D}_{\widehat{t}}^{\xi }(S\ast S)\left(\widehat{t}\right)$$

Upon taking the inverse Laplace transform of the left-hand side

$${\mathcal{L}}^{-1}\left\{{\int }_0^{\infty }{e}^{-\lambda \widehat{t}}S\left(\widehat{t}\right)d\widehat{t}-{\int }_0^{\infty }{e}^{-\mu s}S\left(s\right)ds\right\}\left(\widehat{t}\right)=S\left(\widehat{t}\right)-S\left(\widehat{t}\right)=0$$

By Combining the definition of the three-parameter Mittag–Leffler integral operator ${\mathbb{E}}_{\widehat{t}}^{\mu -\nu ,\mu -\xi ,\mu ,-e,-k}S\left(\widehat{t}\right)$, we eliminate the common terms and rearrange the equation to derive the resolvent equation.

It is directly given by the previous definition that $\parallel S\left(\widehat{t}\right)\parallel \le M{e}^{\omega \widehat{t}}$, If there exists another resolvent $\left\{U\left(\widehat{t}\right)\right\}$ generating A, then for all $x_0\left(x\right)\in D\left(A\right)$, $S\left(\widehat{t}\right)x_0\left(x\right)=U\left(\widehat{t}\right)x_0\left(x\right)$, by the uniqueness of the Laplace transform, ${\int }_0^{\infty }{e}^{-\lambda \widehat{t}}S\left(\widehat{t}\right)x_0\left(x\right)d\widehat{t}={\int }_0^{\infty }{e}^{-\lambda \widehat{t}}U\left(\widehat{t}\right)x_0\left(x\right)d\widehat{t}$. Since $D\left(A\right)$ is dense, $S\left(\widehat{t}\right)=U\left(\widehat{t}\right)$ holds for all $\widehat{t}>0$.

4. On the Existence and Uniqueness of Strong Solutions

Based on the spectral theory of the Riemann–Liouville fractional $(\mu ,\nu ,\xi ,e,k)$-resolvent operator, we establish well-posedness conditions for strong solutions via the representation $u(\widehat{t},x)=S\left(\widehat{t}\right)x_0\left(x\right)$. Uniqueness is demonstrated through difference function analysis and Titchmarsh’s theorem. The theoretical framework is illustrated with an anomalous diffusion system involving a fourth-order elliptic operator.

Theorem 3.

Suppose A is a closed linear operator on Banach space $X=L^2(\Omega )$, generating an exponentially bounded Riemann–Liouville fractional $(\mu ,\nu ,\xi ,e,k)$-resolvent family ${\left\{S\left(\widehat{t}\right)\right\}}_{\widehat{t}>0}$ with parameter constraints $0<\xi <\nu <\mu \le 1$ and damping coefficients $e,k\in \mathbb{R}$. Then the following conclusions hold for the $RLAFC{P}_{x_0\left(x\right),k}$ with three-parameter damping terms:

• $\left(3.1\right)$ For every initial state $x_0\left(x\right)\in D\left(A\right)$ and $x\in \Omega$, the problem admits a unique strong solution $u(\widehat{t},x)=S(\widehat{t},x)x_0\left(x\right)$, which fulfills $u\in C\left(\right(0,\infty );D(A\left)\right)$;
• $\left(3.2\right)$ The operator A commutes with the resolvent family ${\left\{S\left(\widehat{t}\right)\right\}}_{\widehat{t}>0}$, $AS\left(\widehat{t}\right)x_0\left(x\right)=S\left(\widehat{t}\right)A{x}_0\left(x\right)$ for all $\widehat{t}>0$, $x_0\left(x\right)\in D\left(A\right)$ and $x\in \Omega$. Moreover, the fractional derivatives of the strong solution satisfy the following operational relation:

  $$D_{\widehat{t}}^{\mu }S\left(\widehat{t}\right)x_0\left(x\right)+e{D}_{\widehat{t}}^{\nu }S\left(\widehat{t}\right)x_0\left(x\right)+k{D}_{\widehat{t}}^{\xi }S\left(\widehat{t}\right)x_0\left(x\right)=AS\left(\widehat{t}\right)x_0\left(x\right);$$

  (14)
• $\left(3.3\right)$ There exist constants $M\ge 1$ and $\omega \ge 0$ such that $\parallel S\left(\widehat{t}\right)\parallel \le M{e}^{\omega \widehat{t}}$ for all $\widehat{t}>0$

Then, for any initial state $x_0\left(x\right)\in D\left(A\right)$ and $x\in \Omega$, the $RLAFC{P}_{x_0,k}$ with three-parameter damping terms admits a unique strong solution $u(\widehat{t},x)=S(\widehat{t},x)x_0\left(x\right)$, and this strong solution $u\in C\left(\right(0,\infty );D(A\left)\right)$.

Proof. 

$\left(3.1\right)$ Deduced from the compatibility condition of the resolvent family, since $x_0\left(x\right)\in D\left(A\right)$ implies $A{x}_0\left(x\right)\in X$ and $S\left(\widehat{t}\right):X\to X$ is a linear operator for all $\widehat{t}>0$, it follows that $S\left(\widehat{t}\right)A{x}_0\left(x\right)\in X$. To establish the strong continuity of $u(\widehat{t},x)=S\left(\widehat{t}\right)x_0\left(x\right)$ on $(0,\infty )$ with respect to the graph norm on $D\left(A\right)$, it suffices to verify the continuity of both $u(\widehat{t},x)$ and $Au(\widehat{t},x)$ for all $x_0\left(x\right)\in D\left(A\right)$ and $x\in \Omega$:

• By the strong continuity of the resolvent family, for every initial state $x_0\left(x\right)\in D\left(A\right)$ and $x\in \Omega$:

  $$\underset{\widehat{t}\to {\widehat{t}}_0}{lim}\parallel u(\widehat{t},x)-u({\widehat{t}}_0,x)\parallel =\underset{\widehat{t}\to {\widehat{t}}_0}{lim}\parallel S\left(\widehat{t}\right)x_0\left(x\right)-S\left({\widehat{t}}_0\right)x_0\left(x\right)\parallel =0$$

Similarly, from $Au(\widehat{t},x)=S\left(\widehat{t}\right)A{x}_0\left(x\right)$, we have:

$$\underset{\widehat{t}\to {\widehat{t}}_0}{lim}\parallel Au(\widehat{t},x)-Au({\widehat{t}}_0,x)\parallel =\underset{\widehat{t}\to {\widehat{t}}_0}{lim}\parallel S\left(\widehat{t}\right)A{x}_0\left(x\right)-S\left({\widehat{t}}_0\right)A{x}_0\left(x\right)\parallel =0$$

Therefore, the strong solution satisfies continuity under the graph norm of the operator domain:

$$\underset{\widehat{t}\to {\widehat{t}}_0}{lim}{\parallel u(\widehat{t},x)-u({\widehat{t}}_0,x)\parallel }_{D\left(A\right)}=\underset{\widehat{t}\to {\widehat{t}}_0}{lim}\left(\parallel u(\widehat{t},x)-u({\widehat{t}}_0,x)\parallel +\parallel Au(\widehat{t},x)-Au({\widehat{t}}_0,x)\parallel \right)=0$$

This continuity result aligns with recent findings [18], implying that $u\in C\left(\right(0,\infty );D(A\left)\right)$.

• $\left(3.2\right)$ For the fractional derivative relation, recall the definition of Riemann–Liouville fractional derivative for $\delta \in \{\mu ,\nu ,\xi \}$:

  $$D_{\widehat{t}}^{\delta }u(\widehat{t},x)=D_{\widehat{t}}^{\delta }S\left(\widehat{t}\right)x_0\left(x\right)={\displaystyle \frac{d}{d\widehat{t}}}{J}_{\widehat{t}}^{1-\delta }S\left(\widehat{t}\right)x_0\left(x\right)$$

  For all $\widehat{t}>0$, $x_0\left(x\right)\in D\left(A\right)$ and $x\in \Omega$,

  $$D_{\widehat{t}}^{\mu }S\left(\widehat{t}\right)x_0\left(x\right)+e{D}_{\widehat{t}}^{\nu }S\left(\widehat{t}\right)x_0\left(x\right)+k{D}_{\widehat{t}}^{\xi }S\left(\widehat{t}\right)x_0\left(x\right)=AS\left(\widehat{t}\right)x_0\left(x\right)$$

  Substituting $u(\widehat{t},x)=S\left(\widehat{t}\right)x_0\left(x\right)$ into the above equation, the left-hand side becomes a combination of fractional derivatives of the strong solution:

  $$D_{\widehat{t}}^{\mu }S\left(\widehat{t}\right)x_0\left(x\right)+e{D}_{\widehat{t}}^{\nu }S\left(\widehat{t}\right)x_0\left(x\right)+k{D}_{\widehat{t}}^{\xi }S\left(\widehat{t}\right)x_0\left(x\right)=D_{\widehat{t}}^{\mu }u(\widehat{t},x)+e{D}_{\widehat{t}}^{\nu }u(\widehat{t},x)+k{D}_{\widehat{t}}^{\xi }u(\widehat{t},x)$$

  The right-hand side is $AS\left(\widehat{t}\right)x_0\left(x\right)=Au(\widehat{t},x)$, so the original differential equation holds for all $\widehat{t}>0$, $x_0\left(x\right)\in D\left(A\right)$ and $x\in \Omega$:

  $$AS\left(\widehat{t}\right)x_0\left(x\right)=Au(\widehat{t},x)$$

  Thus, for all $\widehat{t}>0$, the differential equation holds

  $$D_{\widehat{t}}^{\mu }u(\widehat{t},x)+e{D}_{\widehat{t}}^{\nu }u(\widehat{t},x)+k{D}_{\widehat{t}}^{\xi }u(\widehat{t},x)=Au(\widehat{t},x)$$
• $\left(3.3\right)$ Verification of the convolution-type initial condition relies on the properties of fractional integrals:

  $$\underset{\widehat{t}\to 0^{+}}{lim}(G_{1-\mu }\ast u)(\widehat{t},x)=\underset{\widehat{t}\to 0^{+}}{lim}{J}_{\widehat{t}}^{1-\mu }u(\widehat{t},x)$$

  where

  $$J_{\widehat{t}}^{1-\mu }u(\widehat{t},x)={\displaystyle \frac{1}{\Gamma (1-\mu )}}{\int }_0^{\widehat{t}}{(\widehat{t}-\sigma )}^{-\mu }S\left(\sigma \right)x_0\left(x\right)d\sigma$$

  By the strong continuity of the resolvent family $\underset{\sigma \to 0^{+}}{lim}\Gamma \left(\mu \right){\sigma }^{1-\mu }S\left(\sigma \right)x_0\left(x\right)$ = $x_0\left(x\right)$ there exists $ϵ>0$ such that for $\sigma \in (0,ϵ)$

  $$\parallel S\left(\sigma \right)x_0\left(x\right)-{\displaystyle \frac{x_0\left(x\right)}{\Gamma \left(\mu \right){\sigma }^{1-\mu }}}\parallel \le {\displaystyle \frac{C}{{\sigma }^{1-\mu }}}$$

  To validate the interchange of the integral and the limit, we use the dominated convergence theorem, as the integrand is bounded and converges pointwise [5] Chapter 3. Let $\sigma =\widehat{t}s$, then $d\sigma =\widehat{t}ds$, and the integral becomes

  $$J_{\widehat{t}}^{1-\mu }u(\widehat{t},x)={\displaystyle \frac{{\widehat{t}}^{1-\mu }}{\Gamma (1-\mu )}}{\int }_0^1{(1-s)}^{-\mu }S\left(\widehat{t}s\right)x_0\left(x\right)ds$$

  Substitute

  $$S\left(\widehat{t}s\right)x_0\left(x\right)={\displaystyle \frac{x_0\left(x\right)}{\Gamma \left(\mu \right){\left(\widehat{t}s\right)}^{1-\mu }}}+r\left(\widehat{t}s\right)$$

  simplify

  $$J_{\widehat{t}}^{1-\mu }u(\widehat{t},x)={\displaystyle \frac{1}{\Gamma \left(\mu \right)\Gamma (1-\mu )}}{\int }_0^1{s}^{\mu -1}{(1-s)}^{-\mu }{x}_0\left(x\right)ds+{\displaystyle \frac{{\widehat{t}}^{1-\mu }}{\Gamma (1-\mu )}}{\int }_0^1{(1-s)}^{-\mu }r\left(\widehat{t}s\right)ds$$

  By the Beta function formula $B(\mu ,1-\mu )$ = $\Gamma \left(\mu \right)\Gamma (1-\mu )$, the first term equals $x_0\left(x\right)$; Norm of the second term

  $$∥{\displaystyle \frac{{\widehat{t}}^{1-\mu }}{\Gamma (1-\mu )}}{\int }_0^1{(1-s)}^{-\mu }r\left(\widehat{t}s\right)ds∥$$

  $$\le {\displaystyle \frac{C{\widehat{t}}^{1-\alpha }}{\Gamma (1-\mu )}}{\int }_0^1{(1-s)}^{-\mu }{\displaystyle \frac{1}{{\left(\widehat{t}s\right)}^{1-\mu }}}ds={\displaystyle \frac{C}{\Gamma (1-\mu )}}{\int }_0^1{s}^{\mu -1}{(1-s)}^{-\mu }ds={\displaystyle \frac{CB(\mu ,1-\mu )}{\Gamma (1-\mu )}}=C\Gamma \left(\mu \right)$$

  Therefore, as $\widehat{t}\to 0^{+}$

  $$\underset{\widehat{t}\to 0^{+}}{lim}{J}_{\widehat{t}}^{1-\mu }u(\widehat{t},x)=x_0\left(x\right),i.e.,\underset{\widehat{t}\to 0^{+}}{lim}(G_{1-\mu }\ast u)(\widehat{t},x)=x_0\left(x\right)$$

  Take $\delta =\nu$ as an example, the case for $\delta =\xi$ is similar. By semigroup property of fractional integrals

  $$J_{\widehat{t}}^{1-\nu }u(\widehat{t},x)=J_{\widehat{t}}^{\mu -\nu }{J}_{\widehat{t}}^{1-\mu }u(\widehat{t},x)$$

  By local boundedness of $J_{\widehat{t}}^{1-\mu }u(\widehat{t},x)$, we obtain $\underset{\widehat{t}\to 0^{+}}{lim}{J}_{\widehat{t}}^{1-\mu }u(\widehat{t},x)=x_0\left(x\right)$, so there exists $ϵ>0$, such that for $\widehat{t}\in (0,ϵ)$

  $$\parallel J_{\widehat{t}}^{1-\mu }u(\widehat{t},x)\parallel \le \parallel x_0\left(x\right)\parallel +1$$

  For $J_{\widehat{t}}^{\mu -\nu }v(\widehat{t},x)$ = ${\displaystyle \frac{1}{\Gamma (\mu -\nu )}}{\int }_0^{\widehat{t}}{(\widehat{t}-\sigma )}^{(\mu -\nu )-1}v\left(\sigma \right)d\sigma$, if $\parallel v(\widehat{t},x)\parallel \le K$, then

  $$\parallel J_{\widehat{t}}^{\mu -\nu }v(\widehat{t},x)\parallel \le {\displaystyle \frac{K}{\Gamma (\mu -\nu )}}{\int }_0^{\widehat{t}}{(\widehat{t}-\sigma )}^{(\mu -\nu )-1}d\sigma ={\displaystyle \frac{K{\widehat{t}}^{\mu -\nu }}{\Gamma (\mu -\nu +1)}}$$

  As $\widehat{t}\to 0^{+}$, ${\widehat{t}}^{\mu -\nu }\to 0$. so $\underset{\widehat{t}\to 0^{+}}{lim}{J}_{\widehat{t}}^{\mu -\nu }v(\widehat{t},x)=0$ Substitute $v(\widehat{t},x)=J_{\widehat{t}}^{1-\mu }u(\widehat{t},x)$, we get $\underset{\widehat{t}\to 0^{+}}{lim}{J}_{\widehat{t}}^{1-\nu }u(\widehat{t},x)=0$, i.e., $\underset{\widehat{t}\to 0^{+}}{lim}(G_{1-\nu }\ast u)(\widehat{t},x)=0$. □

Theorem 4.

If the following three conditions are satisfied, then the strong solution $u(\widehat{t},x)=S\left(\widehat{t}\right)x_0\left(x\right)$ is unique, and the lower-order initial conditions follow naturally from the higher-order condition:

• $\left(4.1\right)$ For all $\widehat{t}>0$, $x\in \Omega$ and initial state $x_0\left(x\right)\in D\left(A\right)$, $u(\widehat{t},x)=S\left(\widehat{t}\right)x_0\left(x\right)$ is a strong solution to ${\left(RLAFCP\right)}_{x_0,k}$;
• $\left(4.2\right)$ The strong solution depends only on the highest-order convolution initial condition

  $$\underset{\widehat{t}\to 0^{+}}{lim}(G_{1-\mu }\ast u)(\widehat{t},x)=x_0\left(x\right)$$

  The lower-order conditions $\underset{\widehat{t}\to 0^{+}}{lim}(G_{1-\nu }\ast u)(\widehat{t},x)=0$ and $\underset{\widehat{t}\to 0^{+}}{lim}(G_{1-\xi }\ast u)(\widehat{t},x)=0$ are natural corollaries of the strong solution;
• $\left(4.3\right)$ The strong solution satisfies the following strongly continuous integral identity for all $\widehat{t}>0$ and $x\in \Omega$:

  $$J_{\widehat{t}}^{\mu }u(\widehat{t},x)+e{J}_{\widehat{t}}^{\nu }u(\widehat{t},x)+k{J}_{\widehat{t}}^{\xi }u(\widehat{t},x)={\int }_0^{\widehat{t}}Au(\sigma ,x)d\sigma$$

Proof. 

$\left(4.1\right)$ Let $u(\widehat{t},x)$ be any strong solution to ${\left(RLAFCP\right)}_{x_0,k}$, and define the difference function $w(\widehat{t},x)=u(\widehat{t},x)-S\left(\widehat{t}\right)x_0\left(x\right)$ for $\left(\widehat{t}\right)>0$ and $x\in \Omega$. By the regularity requirement of strong solutions:

$$u(\widehat{t},x)\in C((0,\infty );D\left(A\right))$$

combined with the conclusion from the proof of Theorem 4

$$S\left(\widehat{t}\right)x_0\left(x\right)\in C((0,\infty );D\left(A\right))$$

Since $D\left(A\right)$ is a linear space. Thus,

$$w(\widehat{t},x)\in C((0,\infty );D\left(A\right))$$

Since the strong solution $u(\widehat{t},x)$ satisfies the original differential equation and

$$D_{\widehat{t}}^{\mu }S\left(\widehat{t}\right)x_0\left(x\right)+e{D}_{\widehat{t}}^{\nu }S\left(\widehat{t}\right)x_0\left(x\right)+k{D}_{\widehat{t}}^{\xi }S\left(\widehat{t}\right)x_0\left(x\right)=AS\left(\widehat{t}\right)x_0\left(x\right)$$

subtracting the two equations gives

$$D_{\widehat{t}}^{\mu }w(\widehat{t},x)+e{D}_{\widehat{t}}^{\nu }w(\widehat{t},x)+k{D}_{\widehat{t}}^{\xi }w(\widehat{t},x)=Aw(\widehat{t},x)$$

Combined with the initial conditions satisfied by the strong solution $u(\widehat{t},x)$ and those satisfied by $S\left(\widehat{t}\right)x_0\left(x\right)$

$$\underset{\widehat{t}\to 0^{+}}{lim}(G_{1-\mu }\ast S(·)x_0\left(x\right))(\widehat{t},x)=x_0\left(x\right)$$

with all low-order conditions being 0, subtracting the two sets of conditions yields the following:

$$\underset{\widehat{t}\to 0^{+}}{lim}\left(G_{1-\mu }\ast w\right)(\widehat{t},x)=0,\underset{\widehat{t}\to 0^{+}}{lim}\left(G_{1-\nu }\ast w\right)(\widehat{t},x)=0,\underset{\widehat{t}\to 0^{+}}{lim}\left(G_{1-\xi }\ast w\right)(\widehat{t},x)=0$$

$\left(4.2\right)$ Apply the fractional integral $J_{\widehat{t}}^{\mu }$ to both sides of the homogeneous differential equation satisfied by $w(\widehat{t},x)$:

$$J_{\widehat{t}}^{\mu }{D}_{\widehat{t}}^{\mu }w(\widehat{t},x)+e{J}_{\widehat{t}}^{\mu }{D}_{\widehat{t}}^{\nu }w(\widehat{t},x)+k{J}_{\widehat{t}}^{\mu }{D}_{\widehat{t}}^{\xi }w(\widehat{t},x)=w(\widehat{t},x)+e{J}_{\widehat{t}}^{\mu -\nu }w(\widehat{t},x)+k{J}_{\widehat{t}}^{\mu -\xi }w(\widehat{t},x)$$

Simplification of the left-hand side, For any $\delta \in \{\mu ,\nu ,\xi \}$, from

$$J_{\widehat{t}}^{\mu }{D}_{\widehat{t}}^{\delta }w(\widehat{t},x)=J_{\widehat{t}}^{\mu -\delta }{J}_{\widehat{t}}^{\delta }{D}_{\widehat{t}}^{\delta }w(\widehat{t},x)$$

and

$$\underset{\widehat{t}\to 0^{+}}{lim}\left(G_{1-\delta }\ast w\right)(\widehat{t},x)=0=\underset{\widehat{t}\to 0^{+}}{lim}{J}_{\widehat{t}}^{1-\delta }w(\widehat{t},x)=0$$

we obtain

$$J_{\widehat{t}}^{\delta }{D}_{\widehat{t}}^{\delta }w(\widehat{t},x)=w(\widehat{t},x)-\underset{\widehat{t}\to 0^{+}}{lim}{J}_{\widehat{t}}^{1-\delta }w(\widehat{t},x)·G_{\delta }\left(\widehat{t}\right)=w(\widehat{t},x)$$

Thus, the left-hand side simplifies to the following:

$$J_{\widehat{t}}^{\mu }{D}_{\widehat{t}}^{\mu }w(\widehat{t},x)+e{J}_{\widehat{t}}^{\mu }{D}_{\widehat{t}}^{\nu }w(\widehat{t},x)+k{J}_{\widehat{t}}^{\mu }{D}_{\widehat{t}}^{\xi }w(\widehat{t},x)=w(\widehat{t},x)+e{J}_{\widehat{t}}^{\mu -\nu }w(\widehat{t},x)+k{J}_{\widehat{t}}^{\mu -\xi }w(\widehat{t},x)$$

Simplification of the right-hand side, since $w(\widehat{t},x)\in C((0,\infty );D\left(A\right))$ and A is a closed linear operator, the commutativity of A with fractional integrals implies $J_{\widehat{t}}^{\mu }Aw(\widehat{t},x)=A{J}_{\widehat{t}}^{\mu }w(\widehat{t},x)$, so the right-hand side can be expressed as follows:

$$J_{\widehat{t}}^{\mu }Aw(\widehat{t},x)=A{J}_{\widehat{t}}^{\mu }w(\widehat{t},x)$$

Combining both sides, we get the following:

$$w(\widehat{t},x)+e{J}_{\widehat{t}}^{\mu -\nu }w(\widehat{t},x)+k{J}_{\widehat{t}}^{\mu -\xi }w(\widehat{t},x)=A{J}_{\widehat{t}}^{\mu }w(\widehat{t},x)$$

(15)

$\left(4.3\right)$ Multiply both sides of Equation (15) by $S(\widehat{t}-s)$, and integrate with respect to s from 0 to $\widehat{t}$

$${\int }_0^{\widehat{t}}S(\widehat{t}-s)w(s,x)ds+e{\int }_0^{\widehat{t}}S(\widehat{t}-s)J_s^{\mu -\nu }w(s,x)ds+k{\int }_0^{\widehat{t}}S(\widehat{t}-s)J_s^{\mu -\xi }w(s,x)ds={\int }_0^{\widehat{t}}S(\widehat{t}-s)Aw(s,x)ds$$

Utilizing the integral semigroup property of the resolvent family

$${\int }_0^{\widehat{t}}S(\widehat{t}-s)J_s^{\delta }w(s,x)ds=J_{\widehat{t}}^{\delta }{\int }_0^{\widehat{t}}S(\widehat{t}-s)w(s,x)ds$$

Thus, the left-hand side can be simplified to the following:

$${\int }_0^{\widehat{t}}S(\widehat{t}-s)w(s,x)ds+e{J}_{\widehat{t}}^{\mu -\nu }{\int }_0^{\widehat{t}}S(\widehat{t}-s)w(s,x)ds+k{J}_{\widehat{t}}^{\mu -\xi }{\int }_0^{\widehat{t}}S(\widehat{t}-s)w(s,x)ds$$

Simplifying the right-hand side: since $S(\widehat{t}-s)Aw(s,x)=AS(\widehat{t}-s)w(s,x)$ and the closed linear operator A commutes with the integral operation, the right-hand side can be rewritten as follows:

$$A{\int }_0^{\widehat{t}}S(\widehat{t}-s)w(s,x)ds$$

Let $h(\widehat{t},x)={\int }_0^{\widehat{t}}S(\widehat{t}-s)w(s,x)ds$ to obtain the homogeneous equation, Combining the above results, we get the following:

$$h(\widehat{t},x)+e{J}_{\widehat{t}}^{\mu -\nu }h(\widehat{t},x)+k{J}_{\widehat{t}}^{\mu -\xi }h(\widehat{t},x)=Ah(\widehat{t},x)$$

(16)

Meanwhile, the strongly continuity of the resolvent family ${\left\{S\left(\widehat{t}\right)\right\}}_{\widehat{t}>0}$ combined with the continuity of $w(s,x)$ ensures that $h\left(\widehat{t}\right)$ belongs to the space $C\left(\right(0,\infty );X)$, and satisfies $\underset{\widehat{t}\to 0^{+}}{lim}h(\widehat{t},x)=0$.

By the characteristic property of the resolvent family, $S\left(\widehat{t}\right)x$ is the unique solution to the homogeneous Equation (16). Since $h(\widehat{t},x)$ satisfies the same equation and fulfills the zero initial condition $\underset{\widehat{t}\to 0^{+}}{lim}h(\widehat{t},x)=0$, the only possible solution is the trivial one: $h(\widehat{t},x)=0$ for all $\widehat{t}>0$ and $x\in \Omega$.

By differentiation under the integral sign,

$${\displaystyle \frac{d}{d\widehat{t}}}{\int }_0^{\widehat{t}}S(\widehat{t}-s)w(s,x)ds=S\left(0^{+}\right)w(\widehat{t},x)+{\int }_0^{\widehat{t}}{\displaystyle \frac{d}{d\widehat{t}}}S(\widehat{t}-s)w(s,x)ds$$

since $h(\widehat{t},x)=0$, its derivative is also 0. Moreover the family, $\left\{S\left(\widehat{t}\right)\right\}$ forms an exponentially bounded collection of bounded linear operators as established in [19], which directly implies that $w(\widehat{t},x)=0$ for all $\widehat{t}>0$ and $x\in \Omega$.

Conclusion: Since $w(\widehat{t},x)=u(\widehat{t},x)-S\left(\widehat{t}\right)x_0\left(x\right)=0$ for all $\widehat{t}>0$, it follows that $u(\widehat{t},x)=S\left(\widehat{t}\right)x_0\left(x\right)$, establishing that the strong solution to ${\left(RLAFCP\right)}_{x_0,k}$ is unique. □

Example 1.

Consider an anomalous diffusion system with three-parameter fractional damping [11], subject to Dirichlet-type boundary conditions and regularized initial conditions, taking the following specific form:

$$\left\{\begin{array}{c}{D}_{\widehat{t}}^{\mu }u(\widehat{t},x)+e{D}_{\widehat{t}}^{\nu }u(\widehat{t},x)+k{D}_{\widehat{t}}^{\xi }u(\widehat{t},x)=-k^2{\displaystyle \frac{{\partial }^{4}u(\widehat{t},x)}{\partial x^4}}\hfill \\ u(\widehat{t},0)=0,u^{\prime }(\widehat{t},1)=0,u^{″}(\widehat{t},0)=0,u^{″}(\widehat{t},1)=0,\hfill \\ \underset{\widehat{t}\to 0^{+}}{lim}{J}_{\widehat{t}}^{1-\mu }u(\widehat{t},x)=x_0\left(x\right),\hfill \\ \underset{\widehat{t}\to 0^{+}}{lim}{J}_{\widehat{t}}^{1-\nu }u(\widehat{t},x)=0,\underset{\widehat{t}\to 0^{+}}{lim}{J}_{\widehat{t}}^{1-\xi }u(\widehat{t},x)=0\hfill \end{array}\right.$$

(17)

where the parameters satisfy $0<\xi <\nu <\mu \le 1$, e and k are real constants, $k>0$, $\widehat{t}>0$, and $x\in \Omega =(0,1)$, the state space $X=L^2(0,1)$ and the operator $A={\displaystyle \frac{{\partial }^4}{\partial x^4}}$ with domain $D\left(A\right)=v\in H^4(0,1)\mid v\left(0\right)=v^{\prime }\left(1\right)=0,v^{″}\left(0\right)=v^{″}\left(1\right)=0$, The eigenvalues of A are ${\lambda }_n={\left(n\pi \right)}^4$, and the eigenfunctions are ${\varphi }_n\left(x\right)$ = $sin\left({\displaystyle \frac{(2n-1)\pi x}{2}}\right)$.

This example is not a generic application of fractional diffusion equations, but a targeted verification case aligned with the core innovation of the three-parameter model, addressing the key challenge of two-parameter models in accurately characterizing multi-scale damping in engineering practice.

Specifically, the fourth-order elliptic operator $A={\displaystyle \frac{{\partial }^4}{\partial x^4}}$ precisely corresponds to the vibration damping control of carbon fiber laminates in the aerospace field, where energy dissipation inherently involves three critical scales: one is the low-frequency damping $\left(\mu \right)$ induced by macroscopic bending of the component, which can be partially characterized by two-parameter models; the second is the medium-frequency damping $\left(\nu \right)$ caused by fiber-matrix interface friction, which can only be roughly fitted by two-parameter models; the third is the high-frequency damping $\left(\xi \right)$ arising from microscale fracture precursors and transient interface perturbations, which cannot be effectively distinguished by two-parameter models, these models arbitrarily merge it into medium-frequency damping, thereby leading to distorted physical mechanisms and inadequate modeling precision.

This scenario directly reflects the intrinsic limitation of two-parameter models: when high-frequency dissipation dominates transient responses, their two-scale averaging mechanism fails to capture real physical processes. By targeting this engineering-critical scenario, the example rigorously verifies the necessity and practical value of the three-parameter model for separating and characterizing multi-scale damping, highlighting its superiority over classical two-parameter frameworks.

• $\left(1\right)$ Existence of Strong Solutions

Let $u(\widehat{t},x)={\sum }_{n=1}^{\infty }{u}_n\left(\widehat{t}\right){\varphi }_n\left(x\right)$, and expand the initial function as $x_0\left(x\right)={\sum }_{n=1}^{\infty }{x}_{0n}{\varphi }_n\left(x\right)$. Substituting these expansions into Equation (17), we derive the following fractional differential equation for the coefficient $u_n\left(\widehat{t}\right)$:

$$D_{\widehat{t}}^{\mu }{u}_n+e{D}_{\widehat{t}}^{\nu }{u}_n+k{D}_{\widehat{t}}^{\xi }{u}_n=-k^2{\lambda }_n{u}_n,\underset{\widehat{t}\to 0^{+}}{lim}{J}_{\widehat{t}}^{1-\mu }{u}_n=x_{0n}$$

By the existence theorem of resolvent operators [20] $u_n\left(\widehat{t}\right)=S_n\left(\widehat{t}\right)x_{0n}$, where

$$S_n\left(\widehat{t}\right)={\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\widehat{t}}^{\mu -1}{E}_{\mu -\nu ,\mu -\xi ,\mu }(-e{k}^2{\lambda }_n{\widehat{t}}^{\mu -\nu },-k^2{\lambda }_n{\widehat{t}}^{\mu -\xi })$$

(18)

Since $x_0\left(x\right)\in D\left(A\right)$, we have ${\sum }_{n=1}^{\infty }{\lambda }_n^2{\left|x_{0n}\right|}^2<\infty$. Combined with the decay properties of the Mittag–Leffler function, the series ${\sum }_{n=1}^{\infty }{u}_n\left(\widehat{t}\right){\varphi }_n\left(x\right)$ converges uniformly in $D\left(A\right)$, ensuring that $u\in C\left(\right(0,\infty );D(A\left)\right)$. Thus, the existence of the solution is established.

• $\left(2\right)$ Let $u_1,u_2$ be two strong solutions, and define $w=u_1-u_2$. Expanding $w={\sum }_{n=1}^{\infty }{w}_n\left(\widehat{t}\right){\varphi }_n\left(x\right)$, we find that each coefficient $w_n\left(\widehat{t}\right)$ satisfies the following equation:

  $$D_{\widehat{t}}^{\mu }{w}_n+e{D}_{\widehat{t}}^{\nu }{w}_n+k{D}_{\widehat{t}}^{\xi }{w}_n=-k^2{\lambda }_n{w}_n$$

  $$\underset{\widehat{t}\to 0^{+}}{lim}{J}_{\widehat{t}}^{1-\mu }{w}_n=0,\underset{\widehat{t}\to 0^{+}}{lim}{J}_{\widehat{t}}^{1-\nu }{w}_n=0,\underset{\widehat{t}\to 0^{+}}{lim}{J}_{\widehat{t}}^{1-\xi }{w}_n=0$$

  By the uniqueness of the resolvent operator, $w_n\left(\widehat{t}\right)\equiv 0$, which implies $w\equiv 0$, thereby proving uniqueness [7]. Conclusion: The equation admits a unique strong solution.

  $$u(\widehat{t},x)=\sum _{n=1}^{\infty }{\displaystyle \frac{f_n}{\Gamma \left(\mu \right)}}{\widehat{t}}^{\mu -1}{E}_{\mu -\nu ,\mu -\xi ,\mu }(-e{k}^2{\left(n\pi \right)}^4{\widehat{t}}^{\mu -\nu },-k^2{\left(n\pi \right)}^4{\widehat{t}}^{\mu -\xi })sin\left({\displaystyle \frac{(2n-1)\pi x}{2}}\right)$$

  This extends recent work on fractional viscoelastic models [9].

5. Conclusions

This study established a rigorous theoretical framework for the existence and uniqueness of strong solutions to fractional diffusion equations enhanced with a three-parameter damping term. By generalizing the classical Riemann–Liouville abstract fractional Cauchy problem, the proposed model more effectively captured multi-scale damping phenomena.

Our principal findings were threefold. First, we precisely delineated the admissible parameter range for the additional fractional order $\xi$, ensuring its physical consistency with the initial singularity of the model. Secondly, the introduction of the generalized $(\mu ,\nu ,\xi ,e,k)$-resolvent family and the corresponding Mittag–Leffler function provided a flexible tool for characterizing the solution structure. Thirdly, utilizing this framework, we rigorously proved the well-posedness of the strong solution, with the illustrative example of anomalous diffusion underpinning the theoretical findings.

The significance of this work extended beyond theoretical enrichment. This three-parameter model offered a potent mathematical tool for simulating complex dissipation processes, such as those encountered in multi-frequency viscoelastic vibrations and anomalous diffusion-reaction systems, thereby addressing a limitation of simpler models.

Future research will focus on developing efficient numerical schemes for the presented solutions and extending the analysis to encompass nonlinear fractional diffusion equations with multi-parameter damping.