A key characteristic of the anomalous sub-solution equation is that the solution exhibits algebraic decay rate over long time intervals, which is often refered to the Mittag-Leffler type stability. For a class of power nonlinear sub-diffusion models with variable coefficients, we prove that their solutions have Mittag-Leffler stability when the source functions satisfy natural decay assumptions. That is the solutions have the decay rate $\|u(t)\|_{L^{s}(\Omega)}=O\left( t^{-(\alpha+\beta)/\gamma} \right)$ as $t\rightarrow\infty$, where $\alpha$, $\gamma$ are positive constants, $\beta\in(-\alpha,\infty)$ and $s\in (1,\infty)$. Then we develop the structure preserving algorithm for this type of model. For the complete monotonicity-preserving ($\mathcal{CM}$-preserving) schemes developed by Li and Wang (Commun. Math. Sci., 19(5):1301-1336, 2021), we prove that they satisfy the discrete comparison principle for time fractional differential equations with variable coefficients. Then, by carefully constructing the fine the discrete supsolution and subsolution, we obtain the long time optimal decay rate of the numerical solution $\|u_{n}\|_{L^{s}(\Omega)}=O\left( t_n^{-(\alpha+\beta)/\gamma} \right)$ as $t_{n}\rightarrow\infty$, which is fully agree with the theoretical solution. Finally, we validated the analysis results through numerical experiments.

翻译：反常亚扩散方程的一个关键特征是，其解在长时间区间上呈现代数衰减率，这通常被称为Mittag-Leffler型稳定性。针对一类具有变系数的幂次非线性亚扩散模型，我们证明了当源函数满足自然衰减假设时，其解具有Mittag-Leffler稳定性。即解具有衰减率 $\|u(t)\|_{L^{s}(\Omega)}=O\left( t^{-(\alpha+\beta)/\gamma} \right)$ 当 $t\rightarrow\infty$，其中 $\alpha$、$\gamma$ 为正常数，$\beta\in(-\alpha,\infty)$ 且 $s\in (1,\infty)$。随后，我们为此类模型构建了结构保持算法。对于Li和Wang (Commun. Math. Sci., 19(5):1301-1336, 2021) 所发展的完全单调性保持（$\mathcal{CM}$-保持）格式，我们证明了它们满足变系数时间分数阶微分方程的离散比较原理。进而，通过精细构造离散的上解与下解，我们获得了数值解的长时间最优衰减率 $\|u_{n}\|_{L^{s}(\Omega)}=O\left( t_n^{-(\alpha+\beta)/\gamma} \right)$ 当 $t_{n}\rightarrow\infty$，这与理论解完全一致。最后，我们通过数值实验验证了分析结果。