1490 CHAPTER 45. TRACES OF SOBOLEV SPACES

Lemma 45.3.4 Let Λ be the generator of G(t) and let t → g(t) be in C1 (0,∞;A1). Thenthere exists a unique solution to the initial value problem

y′−Λy = g, y(0) = y0 ∈ D(Λ)

and it is given by

y(t) = G(t)y0 +∫ t

0G(t− s)g(s)ds. (45.3.8)

This solution is continuous having continuous derivative and has values in D(Λ).

Proof: First I show the following claim.Claim:

∫ t0 G(t− s)g(s)ds ∈ D(Λ) and

Λ

(∫ t

0G(t− s)g(s)ds

)= G(t)g(0)−g(t)+

∫ t

0G(t− s)g′ (s)ds

Proof of the claim:

1h

(G(h)

∫ t

0G(t− s)g(s)ds−

∫ t

0G(t− s)g(s)ds

)

=1h

(∫ t

0G(t− s+h)g(s)ds−

∫ t

0G(t− s)g(s)ds

)=

1h

(∫ t−h

−hG(t− s)g(s+h)ds−

∫ t

0G(t− s)g(s)ds

)

=1h

∫ 0

−hG(t− s)g(s+h)ds+

∫ t−h

0G(t− s)

g(s+h)−g(s)h

−1h

∫ t

t−hG(t− s)g(s)ds

Using the estimate in Theorem 19.14.3 on Page 577 and the dominated convergence theo-rem the limit as h→ 0 of the above equals

G(t)g(0)−g(t)+∫ t

0G(t− s)g′ (s)ds

which proves the claim.Since y0 ∈ D(Λ) ,

G(t)Λy0 = G(t) limh→0

G(h)y0− y0

h

= limh→0

G(t +h)−G(t)h

y0

= limh→0

G(h)G(t)y0−G(t)y0

h(45.3.9)