34.6. ANOTHER APPROACH 1209

where B′ ∈ L∞ (0,T ;L (W,W ′)) . Here W is a Banach space such that V ⊆W. Also VI ≡Lp (I;V ) and WI ≡ L2 (I;W ).

Now let I = [a,b] and c < a < b < d. Here and in what follows φ n (t) = nφ (nt) whereφ ≥ 0,φ ∈C∞

0 (−1,1) , and∫

φdt = 1. The following proposition is known and the essentialfeatures of its proof may be found in [92]. We give a proof for the convenience of the reader.

Proposition 34.6.1 Suppose D(t) ∈ L (W,W ′) and D(t) = 0 if t /∈ (c,d). Suppose alsothat

D(t) =∫ t

cD′ (s)ds, D′ ∈ L∞

(c,d;L

(W,W ′

)).

For u ∈WI and a−n−1 > c, b+n−1 < d, define

Tnu = (D(u∗φ n))′− ((Du)∗φ n)

′ (34.6.42)

where we let u = 0 off I. Then||Tnu||W ′I → 0 (34.6.43)

Proof: First, we show that ||Tn|| is uniformly bounded. Letting w = 0 off I,

|⟨Tnu,w⟩|=∣∣∣∣∫R⟨D′ (t)

∫R

u(s)φ n (t− s)ds,w(t)⟩dt∣∣∣∣

+

∣∣∣∣∫R⟨∫R(D(t)−D(s))u(s)φ

′n (t− s)ds,w(t)⟩dt

∣∣∣∣≤C ||u||WI

||w||WI+∫R

∫R||D(t)−D(s)|| ||u(s)||n2 ∣∣φ ′ (n(t− s))

∣∣ ||w(t)||dsdt

≤C ||u||WI||w||WI

+∫R

∫ 1

−1

∣∣∣∣∣∣D(t)−D(

t− rn

)∣∣∣∣∣∣ ∣∣∣∣∣∣u(t− rn

)∣∣∣∣∣∣n2 ∣∣φ ′ (r)∣∣ ||w(t)|| 1n

drdt

≤C ||u||WI||w||WI

+C∫ 1

−1

∫R

∣∣∣∣∣∣u(t− rn

)∣∣∣∣∣∣W||w(t)||W dtdr

≤C ||u||WI||w||WI

.

Where C is a positive constant independent of n and u. Thus ||Tn|| is bounded independentof n.

Next let u ∈C∞0 (I;V ) , a dense subset of WI . Then a little computation shows∣∣⟨Tnu,w⟩WI

∣∣≤C (φ)

∫ b

a

∫ 1

−1

∣∣∣∣∣∣D′ (t)−D′(

t− rn

)∣∣∣∣∣∣ ∣∣∣∣∣∣u(t− rn

)∣∣∣∣∣∣W||w(t)||W drdt

+C (φ)∫ b

a

∫ 1

−1

∣∣∣∣∣∣D(t)−D(

t− rn

)∣∣∣∣∣∣ ∣∣∣∣∣∣u′(t− rn

)∣∣∣∣∣∣W||w(t)||W drdt

≡ A+B.