78.3. RELAXED COERCIVITY CONDITION 2673

Then picking η small enough, we obtain CBλ (ω)ηr/2 < ε/2.Both operators εF and A(·,ω) are pseudomonotone as maps from X to P (X ′) where

X is defined in terms of UrI as before. Therefore, the existence of the measurable solutionis obtained.

Denoting with uε the above solution, suppose Luε → Lu weakly in U ′r with Lu =

(Bu)′ ∈ V ′ and uε → u weakly in V and u∗ε → u∗ in V ′,εFuε → 0 strongly in U ′r . Thus,

passing to the limit in 78.3.25 we obtain Lu ∈ V ′ because it equals something in V ′. Wewill show this below by an argument that εFuε → 0 strongly in U ′

r .Written differently, the uε satisfy the following for all v ∈ X .∫ T

0⟨Luε ,v⟩+ ⟨Buε ,v⟩(0)+

∫ T

0⟨u∗ε ,v⟩

+ε

∫ T

0⟨Fuε ,uε⟩=

∫ T

0⟨ f ,v⟩+ ⟨Bv(0) ,u0⟩ (78.3.27)

andBuε (t) = Bu0 +

∫ t

0Luε (s)ds

The weak convergence of Luε implies that Buε (t)→ Bu(t) in U ′. Thus

Bu(t) = Bu0 +∫ t

0Lu(s)ds

and so Bu(0) = Bu0. We will show that there exist suitable subsequences such that the kindof convergence just described will hold.

Using the equation to act on u in 78.3.25 or in 78.3.27, we obtain from the assumedcoercivity condition the following for fixed ω,

12⟨Bu,u⟩(t)− 1

2⟨Bu,u⟩(0)+ ε

∫ t

0||u||rU ds+δ (ω)

∫ t

0∥u∥p

V ds−m(ω)

≤ λ (ω)∫ t

0⟨Bu,u⟩(s)ds+

∫ t

0⟨ f ,u⟩(s)ds (78.3.28)

From Gronwall’s inequality, one obtains an estimate of the form

⟨Bu,u⟩(t)+ ε

∫ T

0||u||rU ds+

∫ T

0||u||pV ds≤C ( f ,ω)

where the constant depends only on the indicated quantities. It follows from this and thedefinition of the duality map F that if uε is the solution to Lemma 78.3.1, then εFuε → 0strongly in U ′

r . Also, the estimates for A and the above estimate implies that Luε is boundedin U ′

r . Thus we have an inequality of the form

⟨Buε ,uε⟩(t)+ ε

∫ T

0||uε ||rU ds+ ||uε ||pV + ||Luε ||U ′r + ||u

∗ε ||V ′ ≤C ( f ,ω)

Of course each of these uε ,u∗ε are measurable into V and U ′ respectively. By densityconsiderations, u∗ε is also measurable into V ′. It follows from Theorem 78.1.3 that there