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

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

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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. If x ∈ D(A) and Ax ∈ D(A) it is in fact a strict solution; if x ∈ DA (α + 1, ∞) then it is a strict solution and it also belongs to C 1 ([0, T ]; X) ∩ B([0, T ]; DA (α + 1, ∞)).

8]. 3. Consider the sectorial operators Ap in the sequence spaces D(Ap ) = {(xn ) ∈ p : (nxn ) ∈ p }, and assume that for every f ∈ C([0, T ]; value x = 0 is a strict one. 1) with initial (i) Use the closed graph theorem to show that the linear operator S : C([0, 1]; p ) → C([0, 1]; D(Ap )), Sf = etA ∗ f is bounded. (ii) Let (en ) be the canonical basis of p and consider a nonzero continuous function g : [0, +∞) → [0, 1] with support contained in [1/2, 1]. Let fn (t) = g(2n (1 − t))e2n ; then fn ∈ C([0, 1]; p ), fn ∞ ≤ 1.

In the following two theorems we prove that, under some regularity conditions on f , the mild solution is strict or classical. In the theorem below we assume time regularity whereas in the next one we assume “space” regularity on f . 4). 1), and there is C > 0 such that u C 1 ([0,T ],X) + u C([0,T ],D(A)) ≤ C( f C α ([0,T ],X) + x D(A) ). 15) ≤ C( f C α ([0,T ];X) + x D(A) + Ax + f (0) DA (α,∞) ). Proof. We are going to show that if x ∈ D(A) then u ∈ C((0, T ]; D(A)), and that if x ∈ D(A) and Ax + f (0) ∈ D(A) then u ∈ C([0, T ]; D(A)).

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