Existência, Unicidade e Decaimento Exponencial das Soluç ... - UFRJ

e de acordo com a análise anterior,
∫ t
‖ϕ(u) − ϕ(v)‖ XT
=
∥ S(t − s)(uu x (s) − vv x (s))ds∥
≤
≤
≤
≤
≤
0
∫ t
0
∫ T
0
∥
XT
‖S(t − s)(uu x (s) − vv x (s))‖ XT
ds
{‖S(t − s)(uu x (s) − vv x (s))‖ L ∞ (0,T ;L 2 (Ω))
+ ‖S(t − s)(uu x (s) − vv x (s))‖ L 2 (0,T ;H 1 0 (Ω))}ds
∫ T
0
{‖uu x (s) − vv x (s)‖ L 2 (Ω)
+c(1 + √ T ) ‖uu x (s) − vv x (s)‖ L 2 (Ω) }ds
∫ T
0
∫ T
0
(1 + c + c √ T ) ‖uu x (s) − vv x (s)‖ L 2 (Ω) ds
c 1 (1 + √ T ) ‖uu x (s) − vv x (s)‖ L 2 (Ω) ds
≤ c 1 (1 + √ T ) ‖uu x (s) − vv x (s)‖ L 1 (0,T ;L 2 (Ω)) .
Então, aplicando a desigualdade triangular e a desigualdade de Hölder,
temos
‖ϕ(u) − ϕ(v)‖ XT
≤ c 1 (1 + √ T ) ‖uu x − vu x + vu x − vv x ‖ L 1 (0,T ;L 2 (Ω))
≤
c 1 (1 + √ T ){‖(u − v)u x ‖ L 1 (0,T ;L 2 (Ω))
+ ‖(u − v)u x ‖ L 1 (0,T ;L 2 (Ω)) }
≤
c 1 (1 + √ T ) ‖u − v‖ L 2 (0,T ;L ∞ (Ω)) ‖u x‖ L 2 (0,T ;L 2 (Ω))
+c 1 (1 + √ T ) ‖u x − v x ‖ L 2 (0,T ;L 2 (Ω)) ‖v‖ L 2 (0,T ;L ∞ (Ω)) .
Recordemos, a desigualdade de interpolação clássica (Gagliardo-Nirenberg)
‖w‖ L ∞ (Ω) ≤ c ‖w‖1/2 L 2 (Ω) ‖w x‖ 1/2
L 2 (Ω) , ∀w ∈ H1 0(Ω). (3.10)
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