58
10 Appendix
sup
0≤t
(10.14)
Proof. The proof exploits the energy conservation
d
dt
1
2
Ω R+1
(∇ϕ(t))
2
+ (ψ(t))
2
d
s x +
Ω R+1
U (ϕ(t))d
s x
= 0.
(10.15)
(The above equation follows from the continuity equation for the energy momentum
densities and the fact that there is no momentum flux through the boundary of Ω R+1 ,
since supp u(t) ⊂ Ω R .) In fact, putting
K (t) ≡
1
2
Ω R+1
[(∇ϕ(t))
2
+ (ϕ(t))
2
+ (ψ(t))
2
]d
s x,
one gets from (10.15)
K (t) = K (0) +
Ω R+1
d
s x[U (ϕ(0)) − U (ϕ(t))] +
t
0
dt
Ω R+1
ϕ(t
)ψ(t
)d
s x.
Now, by using the lower bound condition (−U (ϕ(t)) ≤ α + βϕ(t)
2
) and the inequality ϕψ ≤
1
2
(ϕ
2
+ ψ
2
) ≤ (ϕ
2
+ ψ
2
+ (∇ϕ)
2
), we have
K (t) ≤ (K (0) + const) + (2γ + 1)
t
0
dt
K (t
).
Then, by Gronwall’s lemma one has
K (t) ≤ (K (0) + const)e
(2γ+1)|t|
which implies (10.14).
Now, we can sketch the proof of Theorem 4.1. Any u(¯ t), 0 ≤ t < T , defined by
the solution for small times, for initial data of compact support, can be chosen as
initial data for the equation
u(t) = W (t − ¯
t) u(¯ t) +
t
¯
t
W (t − s) f (u(s)) ds,
equivalently for the equation
v(τ ) = W (τ )v 0 +
τ
0
W (τ − s) f (v(s)) ds,
(10.16)
10 Appendix
sup
0≤t
(10.14)
Proof. The proof exploits the energy conservation
d
dt
1
2
Ω R+1
(∇ϕ(t))
2
+ (ψ(t))
2
d
s x +
Ω R+1
U (ϕ(t))d
s x
= 0.
(10.15)
(The above equation follows from the continuity equation for the energy momentum
densities and the fact that there is no momentum flux through the boundary of Ω R+1 ,
since supp u(t) ⊂ Ω R .) In fact, putting
K (t) ≡
1
2
Ω R+1
[(∇ϕ(t))
2
+ (ϕ(t))
2
+ (ψ(t))
2
]d
s x,
one gets from (10.15)
K (t) = K (0) +
Ω R+1
d
s x[U (ϕ(0)) − U (ϕ(t))] +
t
0
dt
Ω R+1
ϕ(t
)ψ(t
)d
s x.
Now, by using the lower bound condition (−U (ϕ(t)) ≤ α + βϕ(t)
2
) and the inequality ϕψ ≤
1
2
(ϕ
2
+ ψ
2
) ≤ (ϕ
2
+ ψ
2
+ (∇ϕ)
2
), we have
K (t) ≤ (K (0) + const) + (2γ + 1)
t
0
dt
K (t
).
Then, by Gronwall’s lemma one has
K (t) ≤ (K (0) + const)e
(2γ+1)|t|
which implies (10.14).
Now, we can sketch the proof of Theorem 4.1. Any u(¯ t), 0 ≤ t < T , defined by
the solution for small times, for initial data of compact support, can be chosen as
initial data for the equation
u(t) = W (t − ¯
t) u(¯ t) +
t
¯
t
W (t − s) f (u(s)) ds,
equivalently for the equation
v(τ ) = W (τ )v 0 +
τ
0
W (τ − s) f (v(s)) ds,
(10.16)