Existence of periodic and S‐asymptotically periodic solutions to fractional diffusion equations with analytic semigroups

Abstract

In this paper, we study periodic and S‐asymptotically periodic solutions for fractional diffusion equations (FDE). As we all know, there is no exact periodic solution to differential equations with Caputo or Riemann‐Liouville fractional derivatives. Even so, in this paper, periodic (S‐asymptotically periodic) mild or classical solutions for FDE with Weyl‐Liouville fractional derivatives could be obtained in various fractional power spaces. In addition, a numerical simulation example and a specific example of fractional diffusion equation are given to verify the main theoretical results.

Existence and asymptotic behavior of square-mean S-asymptotically periodic solutions of fractional stochastic evolution equations

“…In particular, Ren et al [33] have proved that there is no nontrivial periodic solution for a class of fractional evolution equations with the Caputo fractional derivative with lower limit at 0. Therefore, since Henríquez et al [20] first studied the S-asymptotic periodic function in Banach space, many scholars have begun to pay attention to the S-asymptotic periodic solution of the fractional evolution equation( such as [12,24,27,31,33]). In particular, in recent years, we have also considered a class of S-asymptotically periodic problems for fractional evolution equations in [24], and obtained some existence results and global asymptotic behavior of S-asymptotically periodic mild solution.…”