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

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18). Then, DA (α, ∞) = Cb2α (Rn ), with equivalence of the norms. 19). Then, DA (α, ∞) = C02α (Ω) := {f ∈ C 2α (Ω) : f|∂Ω = 0}, with equivalence of the norms. 20). Then  2α if 0 < α < 1/2,  C (Ω), DA (α, ∞) =  {f ∈ C 2α (Ω) : Bf|∂Ω = 0}, if 1/2 < α < 1, with equivalence of the norms. 48 Chapter 3. 10(ii) show that, for any α ∈ (0, 1), the operator A : {u ∈ C 2+2α ([0, 1]) : u(0) = u (0) = u(1) = u (1) = 0} → C02α ([0, 1]), Au = u is sectorial. 3(5) which states that the realization of the second order derivative with Dirichlet boundary condition in C 2α ([0, 1]) is not sectorial.

Let H be a Hilbert space and A : D(A) ⊂ H → H be a linear operator. 3) is equivalent to Re Ax, x ≤ 0 for any x ∈ D(A). 3. 2, so that A2 is sectorial in Cb (R). (b) Prove that for each a, b ∈ R a suitable realization of the operator A defined by (Af )(x) = x2 f (x) + axf (x) + bf (x) is sectorial. [Hint. 3. Second method: determine explicitly the resolvent operator using the changes of variables x = et and x = −et ]. 7) where f is a given function in X, X = Lp (RN ), 1 ≤ p < +∞, or X = Cb (RN ).

Notice that both u1 (t) and u2 (t) belong to D(A) for t > 0. Concerning u1 (t), the estimate Ae(t−s)A (f (s) − f (t)) ≤ M1 (t − s)α [f ]C α t−s implies that the function s → e(t−s)A (f (s)−f (t)) is integrable with values in D(A), whence u1 (t) ∈ D(A) for every t ∈ (0, T ] (the same holds, of course, for t = 0 as well). 6(ii). 17)    (ii) Au2 (t) = AetA x + (etA − I)f (t), 0 < t ≤ T. 17)(ii) holds for t = 0, too. Let us show that Au1 is H¨older continuous in [0, T ]. 6). Hence, Au1 is α-H¨older continuous in [0, T ].

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


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