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

Abstract

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 any discretisation of this problem has previously been proved. In the present paper it is shown that for the class of complete monotonicity-preserving ($\mathcal{CM}$-preserving) schemes (which includes the L1 and Grünwald-Letnikov schemes) on uniform meshes $\{t_n:=nh{\}}_{n=0}^{\infty }$, the discrete solution also has $O(t_n^{-\alpha /\gamma })$ decay as $t_n\to \infty$. This result is then extended to $\mathcal{CM}$-preserving discretisations of certain time-fractional nonlinear subdiffusion problems such as the time-fractional porous media and $p$-Laplace equations. For the L1 scheme, the $O(t_n^{-\alpha /\gamma })$ decay result is shown to remain valid on a very general class of nonuniform meshes. Our analysis uses a discrete comparison principle with discrete subsolutions and supersolutions that are carefully constructed to give tight bounds on the discrete solution. Numerical experiments are provided to confirm our theoretical analysis.