2626 CHAPTER 77. STOCHASTIC INCLUSIONS

Let ω→ u0 (ω) be F measurable into W . Also let ω→ f (·,ω) be F measurable intoV ′, ω → B(ω) measurable into L (W,W ′). Now let uh for h > 0 and small, be the uniquesolution to the initial value problem

(B(ω)uh (·,ω))′+ εFuh (·,ω) = f (·,ω)−u∗h (·,ω) in U ′, (77.5.37)Buh (0,ω) = Bu0 (ω)

where u∗h ∈ A(τhu,ω) is a F measurable selection into V ′. Since F is monotone boundedand hemicontinuous, there is no problem with it being pseudomonotone from XrI to X ′rI .Such a solution exists on [0,h] by the above reasoning. Let this solution be denoted byu1. Then use it to define a solution to the evolution equation on [0,2h] called u2. Byuniqueness, these coincide on [0,h]. Then use u2 to extend to a solution on [0,3h] called u3.Then u3 = u2 on [0,2h]. Continue this way to obtain a solution valid on [0,T ]. By Lemma77.4.3, this solution may be assumed to be measurable into U ′. One gets this by using thelemma on a succession of intervals [0,h] , [0,2h] , and so forth.

Now acting on uh and suppressing the dependence on ω in most places, it follows fromthe assumed estimates (Note how the assumption on growth was used here.) that

12⟨Buh,uh⟩(T )−

12⟨Bu0,u0⟩+ ε

∫ T

0∥uh∥r

U ds

≤(∫ T

0∥ f∥p′

V ′ ds)1/p′(∫ T

0∥uh∥p

V

)1/p

+∫ T

0

(a+b∥τhuh∥p̂−1

V

)∥uh∥V ds

≤ ∥ f∥p̂′

V ′ +∥uh∥p̂V +∥uh∥p̂

V +aT 1/ p̂′ +b∥uh∥ p̂V (77.5.38)

which is of the form≤C

(∥ f∥p̂′

V ′ ,a(ω)T)+(2+b)∥uh∥p̂

V

Now here is where it is good that p̂ < r.

∥uh∥p̂V ≤

∫ T

0

1δ

δ ∥uh∥p̂U ds

≤(∫ T

0δ

r/p̂ ∥uh∥rU ds

)p̂/r(∫ T

0

1

δr/(r−p̂)

1r/r−p̂)(r−p̂)/r

≤ 1

δr/(r−p̂)

T (r−p̂)/r

r(r− p̂)+

p̂δr/p̂ ∥uh∥r

U

r

Thus this has shown

12⟨Buh,uh⟩(T )−

12⟨Bu0,u0⟩+ ε ∥uh∥r

U

≤ C(∥ f∥p̂′

V ′ ,a(ω)T)+

1

δr/(r−p̂)

T (r−p̂)/r

r(r− p̂)+

p̂δr/p̂ ∥uh∥r

U

r