Abstract. We compute syntomic cohomology of semistable affinoids in terms of cohomology of (ϕ, Γ)-modules which, thanks to work of Fontaine-Herr, Andreatta-Iovita, and Kedlaya-Liu, is known to compute Galois cohomology of these affinoids. For a semistable scheme over a mixed characteristic local ring this implies a comparison isomorphism, up to some universal constants, between truncated sheaves of p-adic nearby cycles and syntomic cohomology sheaves. This generalizes the comparison results of Kato, Kurihara, and Tsuji for small Tate twists (where no constants are necessary) as well as the comparison result of Tsuji that holds over the algebraic closure of the field. As an application, we combine this local comparison isomorphism with the theory of finite dimensional Banach Spaces and finiteness of etale cohomology of rigid analytic spaces proved by Scholze to prove a Semistable conjecture for formal schemes with semistable reduction. Let O K be a complete discrete valuation ring with fraction field K of characteristic 0 and with perfect residue field k of characteristic p. Let O F = W (k) and F = O F [ 1 p ] so that K is a totally ramified extension of F ; let e = [K : F ] be the absolute ramification index of K. Let O K denote the integral closure of O K in K. Set G K = Gal(K/K), and let ϕ = ϕ W (k) be the absolute Frobenius on W (k). For a log-scheme X over O K , X n will denote its reduction mod p n , X 0 will denote its special fiber.1.1. Statement of the main results.1.1.1. The Fontaine-Messing map. Let X be a fine and saturated log-scheme log-smooth over O K equipped with the log-structure coming from the closed point. Denote by X tr the locus where the log-structure is trivial. This is an open dense subset of the generic fiber of X. For r ≥ 0, let S n (r) X be the (log) syntomic sheaf modulo p n on X 0,ét . It can be thought of as a derived Frobenius and filtration eigenspace of crystalline cohomology or as a relative Fontaine functor. , Kato [36] have constructed period morphisms (i : X 0 ֒→ X, j : X tr ֒→ X) α FM r,n : S n (r) X → i * Rj * Z/p n (r) ′ Xtr , r ≥ 0. from logarithmic syntomic cohomology to logarithmic p-adic nearby cycles. Here we set Z p (r) ′ := 1 p a(r) Z p (r), for r = (p − 1)a(r) + b(r), 0 ≤ b(r) ≤ p − 1. Assume now that X has semistable reduction over O K or is a base change of a scheme with semistable reduction over the ring of integers of a subfield of K. That is, locally, X can be written as Spec(A) for a ring Aétale overIf we put D := {X a+b+1 · · · X d = 0} ⊂ Spec(A) then the log-structure on Spec A is associated to the special fiber and to the divisor D. We have Spec(A) tr = Spec(A K ) \ D K . We prove in this paper that the Fontaine-Messing period map α FM r,n , after a suitable truncation, is essentially a quasi-isomorphism. More precisely, we prove the following theorem. for a universal constant N (not depending on p, X, K, n or r) and a constant c p depending only on p (and d if p = 2).(ii) In general, the kernel and cokernel of this map are annihilated by p N for an int...