Theoretical and numerical investigation of long-time behaviors for time-fractional Fisher equations

In this paper, we deal with the theoretical and numerical analysis of the long-time behavior for nonlinear time-fractional Fisher equations (TFFEs). With the Wirtinger-type inequality, the theoretical asymptotical stability of the zero solution in the sense of the -norm is shown for some smaller growth rate. Numerically, we implement the scheme to approximate the Caputo fractional derivative of order and the finite difference method for the discretization of the space derivative. Meanwhile, a linearly implicit technique is introduced for the nonlinear logistical function. With the investigation on the solvability and positivity of the numerical solutions unconditionally, it is shown that one solution bounds the numerical solutions above. Further, we investigate that the numerical approach is capable of accurately capturing the long-time behavior of the original problems without any step-size restrictions. Finally, numerical examples that are in line with these results are presented to support the theoretical approach, including numerical simulations for large values of.