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Extra resources for Analytic semigroups and reaction-diffusion problems
By Young’s inequality, T (t)f p ≤ f t > 0, 1 ≤ p ≤ +∞. 9) Since Gt and all its derivatives belong to C ∞ (RN ) ∩ Lp (RN ), 1 ≤ p ≤ +∞, it readily follows that the function u(t, x) := (T (t)f )(x) belongs to C ∞ ((0, +∞) × RN ), because we can differentiate under the integral sign. Since ∂Gt /∂t = ∆Gt , then u solves the heat equation in (0, +∞) × RN . Let us show that T (t)f → f in X as t → 0+ if f ∈ Lp (RN ) or f ∈ BU C(RN ). If f ∈ Lp (RN ) we have T (t)f − f p p p Gt (y)f (x − y)dy − f (x) dx = RN RN p Gt (y)[f (x − y) − f (x)]dy dx = RN RN G1 (v)[f (x − = RN G1 (v)|f (x − = p tv) − f (x)]dv dx RN ≤ RN √ √ tv) − f (x)|p dv dx RN |f (x − G1 (v) RN √ tv) − f (x)|p dx dv.
B Conversely, let f ∈ DA (α, ∞). Then, for every t > 0 we have |f (x) − f (y)| ≤ |T (t)f (x) − f (x)| + |T (t)f (x) − T (t)f (y)| + |T (t)f (y) − f (y)| ≤ 2[[f ]]DA (α,∞) tα + |DT (t)f | ∞ |x − y|. 12)(a) is not sufficient for this purpose. To get a better estimate we use the equality n T (n)f − T (t)f = AT (s)f ds, 0 < t < n, t that implies, for each i = 1, . . , N , n Di T (n)f − Di T (t)f = Di AT (s)f ds, 0 < t < n. 12)(b). 14), to get +∞ Di T (t)f = − Di AT (s)f ds, t > 0, t and +∞ Di T (t)f ∞ ≤ f DA (α,∞) t C s3/2−α ds = C(α) f t1/2−α DA (α,∞) .
Prove that Cb1 (R) is of class J1/4 between Cb (R) and Cb4 (R). 4. 6, show that DA (α, ∞) is of class Jα/θ between X and DA (θ, ∞), for every θ ∈ (α, 1). (b) Show that any space of class Jα between X and D(A) is of class Jα/θ between X and DA (θ, ∞), for every θ ∈ (α, 1). (c) Using (a), prove that any function which is continuous with values in X and bounded with values in DA (θ, ∞) in an interval [a, b], is also continuous with values in DA (α, ∞) in [a, b], for α < θ. 5. Prove that for every θ ∈ (0, 1) there is C = C(θ) > 0 such that Di ϕ ∞ Dij ϕ ≤ C( ϕ ∞ (1−θ)/2 ( Cb2+θ (RN ) ) ≤ C( ϕ ϕ 1−θ/2 ( Cb2+θ (RN ) ) (1+θ)/2 , Cbθ (RN ) ) ϕ θ/2 , Cbθ (RN ) ) for every ϕ ∈ Cb2+θ (RN ), i, j = 1, .