By Lorenzi L., Lunardi A., Metafune G., Pallara D.

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N. 16). 12)(b), for each ξ ∈ (0, 1) we get +∞ ξ 1−α AT (ξ)Dij f ∞ = ξ 1−α e−λt AT (ξ + t/2)Dij T (t/2)g dt 0 ∞ +∞ ≤ g DA (α,∞) ξ 1−α 0 +∞ = g DA (α,∞) 0 ck dt (ξ + t/2)(t/2)1−α 2ck ds. 17) Therefore, all the second order derivatives of f are in DA (α, ∞) = Cb2α (RN ), their Cb2α norm is bounded by C g α ≤ C(λ f α + ∆f α ) ≤ max{λC, C} f DA (1+α,∞) , and the statement follows. 8 The case α = 1/2 is more delicate. In fact, the inclusion Lip(RN ) ⊂ DA (1/2, ∞) follows as in the first part of the proof, but it is strict.

25) Proof. 25) holds. For 0 ≤ t ≤ T , v(t) belongs to D(A), and, denoting by |f | the norm of f in B([0, T ]; DA (α, ∞)) t (t − s)α−1 ds ≤ Av(t) ≤ M1,α |f | 0 T α M1,α |f |. 27) so that Av is bounded with values in DA (α, ∞). 1. 28) hence Av is α-H¨older continuous in [0, T ]. 28). 6. 4). Then, u ∈ C 1 ((0, T ]; X) ∩ C((0, T ]; D(A)), and u ∈ B([ε, T ]; DA (α + 1, ∞)) for every ε ∈ (0, T ). 29) ≤ C( f B([0,T ];DA (α,∞)) + x DA (α,∞) ). Proof. Let us write u(t) = etA x + (etA ∗ f )(t). If x ∈ D(A), the function t → etA x is the classical solution of w = Aw, t > 0, w(0) = x.

11). 10), since Cb2+2α (ReN ) ⊂ D(A). To prove the other embedding we have to show that the functions in DA (1 + α, ∞) have second order derivatives belonging to Cb2α (RN ). Fix any λ > 0 and any f ∈ DA (1 + α, ∞). 22) we have +∞ f (x) = e−λt (T (t)g)(x)dt, x ∈ RN . 0 We can differentiate twice with respect to x, because for each i, j = 1, . . , N , both e−λt Di T (t)g ∞ and e−λt Dij T (t)g ∞ are integrable in (0, +∞). 12)(a) we get Dij T (t)g ∞ = Dj T (t/2)Di T (t/2)g c C(α) g 1/2 (t/2) (t/2)1/2−α k g DA (α,∞) .

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Analytic semigroups and reaction-diffusion problems by Lorenzi L., Lunardi A., Metafune G., Pallara D.


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