10.2 The Cauchy Problem for Small Times
57
d(u, v) = sup
0≤t≤T
u(t) − v(t) Ω R+t+1 .
In fact, by using (10.13) (u 0 fixed), the hyperbolic character of W (t) and the local
Lipschitz property of f (u), one has, for 0 ≤ t < T, T small enough,
(Su)(t) − (Sv)(t) Ω R+T +1 ≤
≤ e
t/2
t
0
ds e
s/2 ¯
C(Ω R+T +1 , ρ)u(s) − v(s) Ω R+T +1 ≤
≤ e
t/2 t ¯
C(Ω R+T +1 , ρ)d(u, v)
and S maps E(T, ρ) into itself since
(Su)(t) Ω R+T +1 ≤ e
t/2
1
2
ρ + t ¯
C(Ω R+T +1 , ρ)ρ
≤ ρ.
By Banach theorem on contractions, S has a fixed point which is the required solution
in the interval [0, T ).
In the case in which u 0 does not have a compact support, we introduce a space cutoff
putting
u 0n ≡
χ n ϕ 0
χ n ψ 0
, χ n (x) ∈ C
∞
0 (R
s
),
χ n (x) = 1, if |x| ≤ n, χ n (x) = 0 if |x| ≥ 2n. Then, by the previous argument, (4.6)
has a solution u n (t).
Now, for any sphere Ω R−t , by using the local Lipschitz condition and Gronwall’s
lemma, as in the derivation of (10.12), we get
u n (t) − u m (t) Ω R−t ≤ exp
1
2
+ ¯
C
Ω R , ρ
t
u 0n − u 0m Ω R
and since u 0n converges in X loc to u 0 as n → ∞, also u n converges in X loc and it
converges to the solution of (4.6), with initial data u 0 .
10.3 The Global Cauchy Problem
To prove Theorem 4.1 we start by establishing the following a priori estimate.
Lemma 10.2 If the potential U is such that the local Lipschitz condition and the
lower bound condition are satisfied, then any solution u ∈ C
0
(X loc , [0, T ]) of (4.6)
with supp 0≤t