76.5. REPLACING Φ WITH σ (u) 2571

are monotone hemicontinuous and bounded where

A(ω)u(t)≡ A(t,u(t) ,ω)

and (t,u,ω)→ A(t,u,ω) is progressively measurable. Also suppose for p ≥ 2, the coer-civity, and the boundedness conditions

λ ⟨Bu,u⟩+ ⟨A(t,u,ω) ,u⟩V ≥ δ ∥u∥pV − c(t,ω) (76.4.53)

where c ∈ L1 ([0,T ]×Ω) for all λ large enough. Also,

∥A(t,u,ω)∥V ′ ≤ k∥u∥p−1V + c1/p′ (t,ω) (76.4.54)

also suppose the monotonicity condition for all λ large enough.

⟨(λB+A(ω))(u)− (λB+A(ω))(v) ,u− v⟩ ≥ δ ∥u− v∥2W (76.4.55)

Then if u0 ∈ L2 (Ω,W ) with u0 F0 measurable, there exists a unique solution u(·,ω) ∈ Vω

with u ∈ V (Lp ([0,T ]×Ω,V ) and progressively measurable) such that for ω off a set ofmeasure zero,

Bu(t,ω)−Bu0 (ω)+∫ t

0A(s,u(s,ω) ,ω)ds =

∫ t

0f ds+B

∫ t

0ΦdW.

It is also assumed that V is a reflexive separable real Banach space.

76.5 Replacing Φ With σ (u)It is not hard to include the case where Φ is replaced with a function σ (u) . We make thefollowing assumptions. For each r > 0 there exists λ large enough that

⟨λB(u)+A(u)− (λB(û)+A(û)) ,u− û⟩ ≥ r∥u− û∥2W

Note that in the case where B = I and there is a conventional Gelfand triple, V,H,V ′,this kind of condition is obvious if λ I +A is monotone for some λ . Thus this is not anunreasonable assumption to make although it is stronger than some of the assumptionsused above with the integral given by

∫ t0 ΦdW .

As to σ we make the following assumptions.

(t,u,ω) ∈ [0,T ]×W ×Ω→ σ (t,u,ω) is progressively measurable into W

∥σ (t,u,ω)∥W ≤C+C∥u∥W∥σ (t,u,ω)−σ (t, û,ω)∥L2(Q1/2U,W) ≤ K ∥u− û∥W

That is, it has linear growth and is Lipschitz.Let λ correspond to r where r−∥B∥K2 > 4. Also let T be such that

ĈeλT K2 < 3