2632 CHAPTER 77. STOCHASTIC INCLUSIONS

Theorem 77.5.6 Let the conditions on A hold 77.5.25 - 77.5.28, 77.5.30 - 77.5.34. Also letB satisfy 77.3.6 and assume, if it depends on ω, it is of the form

B(ω) = k (ω)B, k (ω)≥ 0, k measurable

Let u0 be F measurable into W, and let f be product measurable into V ′, f (·,ω) ∈ V ′.Then there exists a solution to the following evolution inclusion

(B(ω)u(·,ω))′+u∗ (·,ω) = f (·,ω) in V ′

B(ω)u(0,ω) = B(ω)u0 (ω) (77.5.57)

where u∗ (·,ω)∈ A(u(·,ω) ,ω). In addition to this, (t,ω)→ u(t,ω) is product measurableinto V and (t,ω)→ u∗ (t,ω) is product measurable into V ′.

In place of the coercivity condition 77.5.28 assume the coercivity condition involvingboth B and A given in 77.5.29. Then

(B(ω)u(·,ω))′+u∗ (·,ω) = f (·,ω) in U ′ (77.5.58)B(ω)u(0,ω) = B(ω)u0 (ω) (77.5.59)

Thus the following holds in V ′

(B(ω)u(·,ω))(t)−B(ω)u0 (ω)+∫ t

0u∗ (·,ω)ds =

∫ t

0f (s,ω)ds

(Bu)′ ∈ V ′

Proof of Theorem 77.5.6: First consider the claim about replacing the coercivity con-dition. Returning to 77.5.50, one obtains by integrating up to t and adding λ

∫ t0 ⟨Buε ,uε⟩ds

to both sides,12⟨Buε ,uε⟩(t)−

12⟨Bu0,u0⟩+ ε

∫ t

0⟨Fuε ,uε⟩ds

+∫ t

0⟨u∗ε ,uε⟩ds+λ

∫ t

0⟨Buε ,uε⟩ds =

∫ t

0⟨ f ,uε⟩ds+λ

∫ t

0⟨Buε ,uε⟩ds (77.5.60)

Then from the estimate 77.5.29,

12⟨Buε ,uε⟩(t)−

12⟨Bu0,u0⟩+ ε

∫ t

0⟨Fuε ,uε⟩ds

+δ (ω)∫ t

0∥uε∥p

V ds−m(ω) =∫ t

0⟨ f ,uε⟩ds+λ

∫ t

0⟨Buε ,uε⟩ds (77.5.61)

From this, it is a routine use of Gronwall’s inequality to obtain the estimate

ε ⟨Fuε ,uε⟩U ′,U +∥uε∥V ≤C (u0, f ,λ ,ω) (77.5.62)

Then the rest of the argument is the same. You obtain the following in U ′.

B(ω)u(t,ω)−B(ω)u0 (ω)+∫ t

0u∗ (·,ω)ds =

∫ t

0f (s,ω)ds