2586 CHAPTER 76. IMPLICIT STOCHASTIC EQUATIONS
In particular, this holds for n = 1. Therefore, adjusting the constant, it follows that∫Ω
⟨Bu1 (t) ,u1 (t)⟩+∫
Ω
∫ T
0∥u1∥p
V dtds≤C
Consequently, there exists a set of measure zero N such that for ω /∈ N,
⟨Bu1 (t) ,u1 (t)⟩+∫ T
0∥u1∥p
V dt ≤C (ω) (76.7.64)
From the integral equation, it follows that, enlarging N by including countably many setsof measure zero, for ω /∈ N
Bun (t)−Bu1 (t)+∫ t
0A(s,un,ω)−A(s,u1,ω)ds
+∫ t
0h(s,ω)Jn (un)−h(s,ω)J1 (u1)ds = 0
Now it is certainly true that |Jn (un)− J1 (un)| ≤ 2. Thus∫ t
0⟨h(s,ω)Jn (un)−h(s,ω)J1 (u1) ,un−u1⟩ds
=∫ t
0⟨h(s,ω)Jn (un)−h(s,ω)J1 (un) ,un−u1⟩ds
+∫ t
0⟨h(s,ω)(J1 (un)− J1 (u1)) ,un−u1⟩ds
≥−2M∫ t
0|un−u1|ds
Therefore, from the Ito formula and for ω /∈ N,
12⟨Bun (t)−Bu1 (t) ,un (t)−u1 (t)⟩+δ
2∫ t
0∥un−u1∥p
V ds
≤∫ t
02M |un−u1|ds≤
(2+
12
∫ t
0|un−u1|2 ds
)M
≤(
2+12
∫ t
0⟨Bun−Bu1,un−u1⟩ds
)M
where M is an upper bound to h. Then by Gronwall’s inequality
12⟨Bun (t)−Bu1 (t) ,un (t)−u1 (t)⟩ ≤ 2MeMT
Hence
12⟨Bun (t)−Bu1 (t) ,un (t)−u1 (t)⟩+δ
2∫ t
0∥un−u1∥p
V ds≤ 2M+T M2eT M