76.3. THE EXISTENCE OF APPROXIMATE SOLUTIONS 2551

In addition to this, it can be assumed that (t,ω)→ u(t,ω) is progressively measurable intoV . That is, for each ω off a set of measure zero, t → u(t,ω) can be modified on a set ofmeasure zero in [0,T ] such that the resulting u is progressively measurable. Then one alsoobtains that u is the unique solution to the integral equation which holds for a.e. ω

Bu(t,ω)−Bu0 +∫ t

0A(s,u(s,ω) ,ω)ds =

∫ t

0f (s,ω)ds+Bq(t,ω) (76.3.22)

Proof: RecallĀ(ω)(t,u)≡ A(t,u+q(t,ω) ,ω)

where q was in V . Therefore, replace this definition of Ā with

Ā(ω)(t,u)≡ A(t,u+q(t,ω)+u0,ω)

Then from Lemma 76.3.5, there exists w ∈ V such that

(Bw)′ (·,ω)+A(·,w(·,ω)+q(·,ω)+u0 (ω) ,ω) = f (·,ω) , Bw(0) = 0

Let u(t,ω) = w(t,ω)+q(t,ω)+u0 (ω) . Then for fixed ω, Bu(0) = Bw(0)+Bu0 = Bu0.Also

(B(u−q))′+A(·,u,ω) = f (·,ω) , Bu(0) = Bu0

Then an integration yields 76.3.22. Uniqueness follows from the above uniqueness as-sumption 76.3.17.

One can easily generalize this using an exponential shift argument.

Corollary 76.3.7 Suppose the situation of the above proposition but that all that is knownis that λB+A is monotone and hemicontinuous on Vω and V for all λ sufficiently large.Then defining

⟨Aλ (t,w,ω) ,v⟩V ′,V ≡⟨

e−λ tA(

t,eλ tw,ω),v⟩

V ′,V

for such λ , it follows that λB+ Aλ is also monotone and hemicontinuous. Then replace thecoercivity, and boundedness conditions above with the following weaker conditions

λ ⟨Bu,u⟩+ ⟨A(t,u,ω) ,u⟩V ≥ δ ∥u∥pV − c(t,ω) (76.3.23)

for all λ large enough.

∥A(t,u,ω)∥V ′ ≤ k∥u∥p−1V + c1/p′ (t,ω) (76.3.24)

where c ∈ L1 ([0,T ]×Ω) , c ≥ 0. Then the conclusion of Proposition 76.3.6 is still valid.There exists a unique u ∈ V such that for a.e. ω,

(Bu−Bq)′ (·,ω)+A(ω)(u(·,ω)) = f (·,ω) , B(u−q)(0) = Bu0 (76.3.25)

Proof: That λB+Aλ is monotone and hemicontinuous follows from the definition.Also, from the above estimates,

λ ⟨Bu,u⟩+ ⟨Aλ (t,u,ω) ,u⟩V ≥ e−2λ t(

λ

⟨B(

eλ tu),eλ tu

⟩+⟨

A(

t,eλ tu,ω),eλ tu

⟩)