In this paper, we study the asymptotic behavior of solutions of semilinear abstract differential equations (*) u′(t) = Au(t) + tnf(t, u(t)), where A is the generator of a C0-semigroup (or group) T(·), f(·, x) ∈ A for each x ∈ X, A is the class of almost periodic, almost automorphic or Levitan almost periodic Banach space valued functions ϕ: ℝ → X and n ∈ {0, 1, 2, ...}. We investigate the linear case when T(·)x is almost periodic for each x ∈ X; and the semilinear case when T(·) is an asymptotically stable C0-semigroup, n = 0 and f(·, x) satisfies a Lipschitz condition. Also, in the linear case, we investigate (*) when ϕ belongs to a Stepanov class Sp-A defined similarly to the case of Sp-almost periodic functions. Under certain conditions, we show that the solutions of (*) belong to Au:= A ∩ BUC(ℝ, X) if n = 0 and to tnAu ⊕ wnC0 (ℝ, X) if n ∈ ℕ, where wn(t) = (1 + |t|)n. The results are new for the case n ∈ ℕ and extend many recent ones in the case n = 0.