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The solution of the nonlinear initial-value problem$\mathcal{D}_{t}^{\alpha}y(t)=-\lambda y(t)^{\gamma}$ for $t>0$ with $y(0)>0$,where $\mathcal{D}_{t}^{\alpha}$ is a Caputo derivative of order $\alpha\in(0,1)$ and $\lambda, \gamma$ are positive parameters, is known to exhibit$O(t^{\alpha/\gamma})$ decay as $t\to\infty$. No corresponding result for anydiscretisation of this problem has previously been proved. In the present paperit is shown that for the class of complete monotonicity-preserving($\mathcal{CM}$-preserving) schemes (which includes the L1 andGr\"unwald-Letnikov schemes) on uniform meshes $\{t_n:=nh\}_{n=0}^\infty$, thediscrete solution also has $O(t_{n}^{-\alpha/\gamma})$ decay as$t_{n}\to\infty$. This result is then extended to $\mathcal{CM}$-preservingdiscretisations of certain time-fractional nonlinear subdiffusion problems suchas the time-fractional porous media and $p$-Laplace equations. For the L1scheme, the $O(t_{n}^{-\alpha/\gamma})$ decay result is shown to remain validon a very general class of nonuniform meshes. Our analysis uses a discretecomparison principle with discrete subsolutions and supersolutions that arecarefully constructed to give tight bounds on the discrete solution. Numericalexperiments are provided to confirm our theoretical analysis.

Optimal long-time decay rate of solutions of complete monotonicity-preserving schemes for nonlinear time-fractional evolutionary equations