Large-Time Upper Decay Estimates for Weakly Coupled Time-Fractional Diffusion Systems
Abstract
We prove conjectured-optimal large-time upper decay estimates for solutions of weakly coupled systems of m≥2 time-fractional diffusion equations. Our result establishes the upper-bound assertion in the conjecture of Li, Liu, and Wada~\cite{LLW26}. The sharpness of these upper exponents for m≥3 remains open in general; see Section~\ref{sec:discussion}. Assume that the diffusion matrices are uniformly elliptic, the coupling coefficients are bounded and may depend on time, and the matrix of L∞-norms of the coupling coefficients satisfies a spectral-radius stability condition. Let α* denote the smallest fractional order among the components with nonzero initial data. Then the L2(Ω)m-norm of the solution is O(t−α*) as t→∞ if α*<1, whereas it is O(t−(1+αm)) if α*=1, where αm is the smallest fractional order occurring in the system. For m=2, the result yields the same upper decay exponents as \cite[Theorem~1.1]{LLW26}; in the absence of self-coupling, the matrix stability condition is weaker than the corresponding small-coupling condition in that work. The case m≥3 is new. Our proof uses a large-time weighted Volterra framework. If α*<1, the weight is a regularly varying power weight; if α*=1, it is constructed by convolution with the resolvent kernel associated with the smallest fractional order. These weights yield the required large-time bounds for all resolvent kernels in the system. In particular, the argument avoids the Laplace-transform and residue-calculus methods used in \cite{LLW26}, which become difficult to implement for systems with three or more components.
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BibTeX
@misc{nguyen2026largetime,
title = {{Large-Time Upper Decay Estimates for Weakly Coupled Time-Fractional Diffusion Systems}},
author = {Huong Nguyen and Thang Nguyen and Ingo Witt},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41993},
eprint = {2610.41993},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41993},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Large-Time Upper Decay Estimates for Weakly Coupled Time-Fractional Diffusion Systems
AU - Huong Nguyen
AU - Thang Nguyen
AU - Ingo Witt
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41993
UR - https://arxiv.org/abs/2610.41993
AB - We prove conjectured-optimal large-time upper decay estimates for solutions of weakly coupled systems of m≥2 time-fractional diffusion equations. Our result establishes the upper-bound assertion in the conjecture of Li, Liu, and Wada~\cite{LLW26}. The sharpness of these upper exponents for m≥3 remains open in general; see Section~\ref{sec:discussion}. Assume that the diffusion matrices are uniformly elliptic, the coupling coefficients are bounded and may depend on time, and the matrix of L^∞-norms of the coupling coefficients satisfies a spectral-radius stability condition. Let α^* denote the smallest fractional order among the components with nonzero initial data. Then the L^2(Ω)^m-norm of the solution is O(t^−α^*) as t→∞ if α^*<1, whereas it is O(t^−(1+α_m)) if α^*=1, where α_m is the smallest fractional order occurring in the system. For m=2, the result yields the same upper decay exponents as \cite[Theorem~1.1]{LLW26}; in the absence of self-coupling, the matrix stability condition is weaker than the corresponding small-coupling condition in that work. The case m≥3 is new. Our proof uses a large-time weighted Volterra framework. If α^*<1, the weight is a regularly varying power weight; if α^*=1, it is constructed by convolution with the resolvent kernel associated with the smallest fractional order. These weights yield the required large-time bounds for all resolvent kernels in the system. In particular, the argument avoids the Laplace-transform and residue-calculus methods used in \cite{LLW26}, which become difficult to implement for systems with three or more components.
ER -