Sharp decay estimates and numerical analysis for weakly coupled systems of two subdiffusion equations

TL;DR

The paper analyzes a two-component weakly coupled time-fractional subdiffusion system with orders $0<\beta<\alpha\le1$, proving a sharp long-time decay dichotomy when the second initial datum vanishes ($v_0\equiv0$): $\|u(t)\|+\|v(t)\|$ decays as $t^{-\alpha}$ for $\alpha<1$ and as $t^{-(1+\beta)}$ for $\alpha=1$. The authors achieve this by combining an energy method with the coercivity of fractional derivatives to reduce the PDE to a coupled fractional ODE system, establishing a maximum principle for the ODEs and deriving precise asymptotics via Laplace transform and residue arguments. They also develop semi-implicit and fully implicit finite-difference schemes with stability guarantees and validate the theoretical decay rates through numerical experiments, including multi-component extensions. The results reveal a distinct acceleration of decay in the coupled setting not present in single fractional diffusion and provide a foundation for future work on multi-component and higher-order fractional systems. These findings have implications for understanding long-time dynamics in anomalous transport models and for designing stable numerical methods for fractional diffusion couplings.

Abstract

This paper investigates the initial-boundary value problem for weakly coupled systems of time-fractional subdiffusion equations with spatially and temporally varying coupling coefficients. By combining the energy method with the coercivity of fractional derivatives, we convert the original partial differential equations into a coupled ordinary differential system. Through Laplace transform and maximum principle arguments, we reveal a dichotomy in decay behavior: When the highest fractional order is less than one, solutions exhibit sublinear decay, whereas systems with the highest order equal to one demonstrate a distinct superlinear decay pattern. This phenomenon fundamentally distinguishes coupled systems from single fractional diffusion equations, where such accelerated superlinear decay never occurs. Numerical experiments employing finite difference methods and implicit discretization schemes validate the theoretical findings.

Figures (7)

• Figure 1: The choice of an integral path $\gamma_{R,\theta_0,\varepsilon}$.
• Figure 2: Long-time decay rates of solutions to the coupled system \ref{['eq-gov2']} with $\alpha=0.9$ and $\beta=0.5$. Left: Case (i) of the initial values. Right: Case (ii) of the initial values.
• Figure 3: Long-time decay rates of solutions to the coupled system \ref{['eq-gov2']} with $\alpha=1.0$ and $\beta=0.5$. Left: Case (i) of the initial values. Right: Case (ii) of the initial values.
• Figure 4: Long-time asymptotic behavior of the solution to the coupled system \ref{['eq-gov2']} with $\alpha=1.0$ and $v_0\equiv0$. Left: $\beta=0.3$. Right: $\beta=0.7$.
• Figure 5: Long-time asymptotic behavior of the solution to the coupled system \ref{['eq-gov3']} with $\alpha=0.9$, $\beta=0.5$ and $\gamma=0.3$. Upper left: Case (i) of the initial values. Upper right: Case (ii) of the initial values. Bottom: Case (iii) of the initial values.

Theorems & Definitions (21)

• Theorem 1.1
• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• Lemma 3.1
• proof