77.5. THE MAIN RESULT 2621

Now suppose the following for the operator A(·,ω) . A(·,ω) : VI → V ′I for each I asubinterval of [0,T ] and

A(·,ω) : VI →P(V ′I) is bounded, (77.5.25)

If, for u ∈ V ,u∗X[0,T̂ ] ∈ A

(uX[0,T̂ ],ω

)for each T̂ in an increasing sequence converging to T, then

u∗ ∈ A(u,ω) (77.5.26)

For some p̂≥ p, assume the specific estimate

sup{∥u∗∥V ′I : u∗ ∈ A(u,ω)

}≤ a(ω)+b(ω)∥u∥p̂−1

VI(77.5.27)

where a(ω) ,b(ω) are nonnegative. Note that the growth could be quadratic in case p = 2.Also assume the coercivity condition:

lim||u||V→∞

u∈Xr

inf{2⟨u∗,u⟩V ′,V + ⟨Bu,u⟩(T ) : u∗ ∈ A(u,ω)}||u||V

= ∞, (77.5.28)

or alternatively the following specific estimate valid for each t ≤ T and for some λ (ω)≥ 0,

inf(∫ t

0⟨u∗,u⟩+λ (ω)⟨Bu,u⟩dt : u∗ ∈ A(u,ω)

)≥ δ (ω)

∫ t

0∥u∥p

V ds−m(ω) (77.5.29)

where m(ω) is some nonnegative constant, δ (ω)> 0. Note that the estimate is a coercivitycondition on λB+A rather than on A but is more specific than 77.5.28.

Let U be a Banach space dense in V and that if ui ⇀ u in VI and u∗i ∈A(ui) with u∗i ⇀ u∗

in V ′I and (Bui)′⇀ (Bu)′ in U ′

rI , ⇀ denoting weak convergence, then if

lim supi→∞

⟨u∗i ,ui−u⟩V ′I ,VI≤ 0

it follows that for all v ∈ VI , there exists u∗(v) ∈ Au such that

lim infi→∞⟨u∗i ,ui− v⟩V ′I ,VI

≥ ⟨u∗ (v) ,u− v⟩V ′I ,VI(77.5.30)

where r > max(p̂,2) , and we replace p with r and I an arbitrary subinterval of the form[0, T̂], T̂ < T, for [0,T ], and U for V where indicated. Here

UrI ≡ Lr (I;U)

Note that we are not assuming A is pseudomonotone, just that it satisfies a similar limitcondition. Typically, this limit condition holds because of a use of the compact embeddingof theorem 77.3.1 or similar result and it does not matter whether U is a small subset of Vas long as it is dense in V .