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By Flaass D.G.

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Now let K > 0 be such that supn∈N ||fn ||∞ ≤ K. 7) into account, we easily deduce that supn∈N ||T (t)fn ||∞ ≤ K(ec0 T ∨ 1) for any t ∈ [0, T ]. 4 then imply that the sequence {T (·)fn } is bounded in C 1+α/2,2+α ([T1 , T2 ] × B(R)). Hence, by the Ascoli-Arzel` a Theorem, there exists a subsequence {Tnk (·)} converging uniformly in [T1 , T2 ] × B(R) to a function v ∈ C 1+α/2,2+α ([T1 , T2 ] × B(R)). Since, T (·)fn converges pointwise to T (·)f in (0, +∞) × RN , we deduce that v = T (·)f and the whole sequence {T (·)fn } converges to T (·)f uniformly in [T1 , T2 ] × B(R).

5) with f ∈ Cb (RN ). , a function u ∈ C([0, +∞) × RN ) ∩ C 1,2 ((0, +∞) × RN )), which is bounded in [0, T ] × RN for any T > 0 and satisfies Dt u, D2 u ∈ α/2,α Cloc ((0, +∞) × RN ). The idea of the proof is similar to that used in the elliptic case. 6)    un (0, x) = f (x), x ∈ B(n), in the ball B(n). 6) admits a unique solution un ∈ 1+α/2,2+α ((0, +∞) × B(n)). 4) and a compactness argument, we prove that we can define a function u : [0, +∞) × RN → R by setting u(t, x) := lim un (t, x), n→+∞ for any t ∈ [0, +∞) and any x ∈ RN .

To complete the proof we must show that u ∈ C([0, +∞) × RN ) and u(0, x) = f (x). For this purpose, we take advantage of the semigroup theory. In particular, we will use the representation formula of solutions to Cauchy-Dirichlet problems in bounded domains through semigroups. Fix M ∈ N and let ϑ be any smooth function such that 0 ≤ ϑ ≤ 1, ϑ ≡ 1 in B(M − 1), ϑ ≡ 0 outside B(M ). For any n > M , let vn = ϑ˜ un . As it is easily seen, the function vn belongs to C([0, +∞) × B(M )) and is the solution of the Cauchy-Dirichlet problem  D v (t, x) − Avn (t, x) = ψn (t, x), t > 0, x ∈ B(M ),    t n vn (t, x) = 0, t > 0, x ∈ ∂B(M ),    vn (0, x) = ϑ(x)f (x), x ∈ B(M ), 12 Chapter 2.

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2-Local subgroups of Fischer groups by Flaass D.G.


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