Multi-term fractional linear equations modeling oxygen subdiffusion through capillaries

1. Problem Statement

The paper investigates the global classical solvability and regularity of a specific class of linear integro-differential equations modeling oxygen transport through capillaries. The governing equation is defined on a space-time cylinder $\Omega_T = \Omega \times (0, T)$ for an unknown function $u(x,t)$:

$$D^{\nu_1}_t (\varrho_1 u) - D^{\nu_2}_t (\varrho_2 u) - L_1 u - K * L_2 u = f$$

Key Components:

• Fractional Derivatives: $D^{\nu_i}_t$ are Caputo fractional derivatives of orders $0 < \nu_2 < \nu_1 \leq 1$.
• Variable Coefficients: $\varrho_1(x,t)$ and $\varrho_2(x,t)$ are time-dependent coefficients, with $\varrho_1 \geq \mu_0 > 0$.
• Elliptic Operators: $L_1$ and $L_2$ are second-order uniform elliptic operators with time-dependent smooth coefficients.
• Memory Terms: $K$ is a summable convolution kernel representing memory effects.
• Boundary Conditions: The problem is analyzed under three types of boundary conditions:
  1. Dirichlet (DBC).
  2. Third-kind (Robin-type, 3BC).
  3. Fractional Dynamic Boundary Condition (FDBC), where the fractional derivative appears on the boundary.

Motivation:
The equation generalizes advanced models of oxygen subdiffusion in tissues. A specific case ($D^{\nu_1}_t C - \tau D^{\nu_2}_t C = \dots$) was previously proposed to model oxygen transport where the net diffusion involves two competing fractional time scales. The authors aim to provide a rigorous mathematical foundation for these models, particularly addressing cases where standard assumptions (like kernel non-negativity) fail.

2. Methodology

The authors employ a sophisticated analytical approach combining parabolic regularization techniques with a priori estimates in fractional Hölder spaces.

A. Functional Framework

The analysis is conducted in fractional Hölder spaces $C^{l+\alpha, \frac{l+\alpha}{2}\nu}(\bar{\Omega}_T)$. These spaces generalize standard parabolic Hölder spaces to accommodate the fractional time derivatives. The authors define norms involving spatial derivatives and fractional time derivatives to control the regularity of solutions.

B. The Regularizer Technique

The core of the proof relies on adapting the regularizer method (originally developed for parabolic equations by Eidelman and others) to this multi-term fractional setting. The strategy involves four main steps:

1. Domain Covering: Constructing a finite covering of the domain $\Omega$ with local coordinate systems (flattening the boundary near $\partial\Omega$).
2. Freezing Coefficients: In small time intervals $[0, \tau]$, the variable coefficients of the operators are "frozen" at specific points to create model problems (constant coefficient equations on $\mathbb{R}^n$ or $\mathbb{R}^n_+$).
3. Construction of the Inverse: Using the known solvability of these model problems, a regularizer operator $R$ is constructed. This operator acts as an approximate inverse to the main differential operator $L$.
4. Perturbation Argument: The authors prove that $L \circ R = I + T_1$ and $R \circ L = I + T_2$, where $T_1$ and $T_2$ are contraction operators for sufficiently small $\tau$. This allows the inversion of $L$ via Neumann series, establishing local solvability.
5. Extension: The local solution is extended step-by-step to the entire interval $[0, T]$.

C. Handling the Negative Kernel

A critical technical challenge is that the difference of two Caputo derivatives, $D^{\nu_1}_t (\varrho_1 u) - D^{\nu_2}_t (\varrho_2 u)$, can be represented as $\frac{\partial}{\partial t}(N * u)$, where the kernel $N(t)$ is not necessarily non-negative.

• Previous literature often required $N(t) \geq 0$ to apply maximum principles or energy estimates.
• Innovation: The authors remove this assumption. By using the regularizer technique and specific estimates on the fractional derivatives of products (Lemmas 5.1 and 5.2), they establish solvability even when the kernel $N$ changes sign (which occurs when $\varrho_1 \leq \varrho_2$ and $t$ is large enough).

3. Key Contributions

1. Generalized Solvability: Proves the existence and uniqueness of a global classical solution for multi-term fractional equations with variable coefficients and memory terms, without assuming the non-negativity of the convolution kernel.
2. Product Rule Handling: Addresses the difficulty that the Leibniz rule does not hold for fractional derivatives. The authors derive specific estimates for terms like $D^\beta_t (\varrho u)$ where $\varrho$ is a variable coefficient, which is essential for the regularizer construction.
3. Dynamic Boundary Conditions: Extends the theory to include Fractional Dynamic Boundary Conditions (FDBC), where the fractional derivative appears explicitly on the boundary, a scenario relevant to physical models of surface interactions.
4. Relaxed Assumptions: The results hold for $\varrho_2$ without sign restrictions, covering cases where the "memory" term might dominate or oppose the primary diffusion term.

4. Main Results

Theorem 4.1 (Main Theorem):
Under assumptions H1–H5 (regarding smoothness of coefficients, compatibility of initial/boundary data, and orders of derivatives), the initial-boundary value problem admits a unique classical solution $u \in C^{2+\alpha, \frac{2+\alpha}{2}\nu_1}(\bar{\Omega}_T)$.

• A Priori Estimates: The solution satisfies specific norm estimates bounded by the norms of the source term $f$, initial data $u_0$, and boundary data $\psi$.
  • For DBC/3BC: $\|u\| + \|D^{\nu_2}_t u\| \leq C (\|f\| + \|u_0\| + \|\psi\|)$.
  • For FDBC: The estimate includes norms of $D^{\nu_1}_t u$ on the boundary.
• Generality: The result applies to $M$-term fractional equations and includes the case where the convolution kernel $K \equiv 0$.

5. Numerical Simulations

The theoretical findings are validated through finite-difference schemes in 1D and 2D domains.

• Scheme: The authors use a Grönwald-Letnikov approximation for fractional derivatives (improved by Richardson extrapolation) and a trapezoidal rule for the convolution integrals.
• Accuracy: The scheme achieves second-order accuracy in space and high accuracy in time.
• Examples:
  • Example 7.2 specifically tests the case where the kernel $N(t)$ changes sign (non-positive), confirming that the numerical solution remains stable and accurate even when the theoretical assumptions of previous works (requiring $N \geq 0$) are violated.
  • Tables 1–6 demonstrate convergence rates and small absolute errors ($\mathcal{E}$) between the exact and numerical solutions for various fractional orders $\nu_1, \nu_2$.

6. Significance

• Mathematical Rigor: This work fills a gap in the theory of fractional partial differential equations (FPDEs) by providing a rigorous framework for multi-term equations with variable coefficients and non-standard kernels.
• Physical Relevance: It validates the mathematical soundness of advanced oxygen transport models used in biomedical engineering. These models often involve complex interactions (subdiffusion, time lags) that result in equations with non-positive kernels.
• Future Directions: The paper establishes a foundation for tackling nonlinear extensions of these equations (e.g., concentration-dependent diffusion) and inverse problems (identifying parameters like diffusion coefficients from experimental data), which are crucial for medical diagnostics and treatment planning.

In summary, the paper successfully bridges the gap between complex physical models of oxygen subdiffusion and rigorous mathematical analysis, overcoming the limitation of kernel non-negativity through a novel application of regularizer techniques.