Some periodic type solutions for stochastic reaction–diffusion equation with cubic nonlinearities

Periodic, almost periodic and almost automorphic solutions for SPDEs with monotone coefficients

“…In particular, X(•) is stationary provided φ, φ i , i = 1, 2, ..., N and B 0 are independent of t. 2. Note that when the noise is additive or of linear form as in (42), the main result of Gao [26] is a special case of the above example for p = 4.…”

“…For infinite dimensional case, Da Prato and Tudor [19] provided the existence of periodic and almost periodic solutions of semilinear stochastic partial differential equations. Later, studies of periodic, almost periodic and almost automorphic solutions to semilinear stochastic differential equations were performed by Bezandry and Diagana [6], Fu and Liu [25], Wang and Liu [58], Chen and Lin [14], Liu and Sun [46], Gao [26], Cheban and Liu [12], Liu and Liu [45], among others. Note that the almost periodic/automorphic solution in [6,25] should be in distribution sense instead of square-mean sense, see Kamenskii et al [37] and Liu and Sun [46] for details.…”

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

“…For these reasons, some scholars have studied the existence of periodic solutions of integer-order diffusion equations. [14][15][16] Thus, the problem that the existence of periodic solutions to fractional diffusion equations is the same as that of integer-order diffusion equations or not is worth studying. Let us note the difference between the integer derivatives and fractional derivatives of periodic functions, 17-19R…”