10.3 The Global Cauchy Problem
59
where v 0 ≡ u(¯ t), v(τ ) ≡ u(τ + ¯
t), and by Lemma 10.2 v 0 Ω R+1 < ρ, ρ > 2L.
Hence, the argument given in Sect. 10.2 can be applied and existence of solutions
for (10.16) can be proved for 0 ≤ τ < T 1 , with T 1 depending only on ρ. Since t can
be chosen as close as we like to T this provides a continuation beyond T .
The existence of solutions for initial data with non-compact support is proved by the
same argument as at the end of Sect 10.2.
10.4 The Non-linear Wave Equation with Driving Term
Theorem 10.2 The equation
δ(t) = W (t)δ 0 + L(t) +
t
0
W (t − s)g(δ(s)) ds,
(10.17)
with L(t), g, δ defined in the proof of Theorem 5.2, L(0) = 0, has a unique solution
δ(t) ∈ C
0
(X, R), X ≡ H
1
(R
s
) ⊕ L
2
(R
s
).
Proof. Uniqueness follows from global Lipschitz continuity by the same argument
of Sect. 10.2, eq. (10.12), since the driving term L(t) cancels. As in Sect. 10.2,
existence of solutions for small times follows by a fixed point argument applied to
the space
E(T, ρ) = {δ ∈ C
0
([0, T ], X ), sup
0≤t≤T
δ(t) X < ρ},
since
(Sδ)(t) ≡ W (t)δ 0 + L(t) +
t
0
W (t − s)g(δ(s))ds
is a contraction on E(T, ρ) for T small enough, δ 0 X < ρ/2. Finally, the continuation beyond T is obtained as in Sect. 10.3, by exploiting the a priori estimate
sup
0≤t δ(t) X ≡ L < ∞,
which follows from energy conservation d E(t)/dt = 0
E(t) =
1
2
d
s x[(∇χ(t))
2
+ (ζ(t) + ψ 0 )
2
] −
χ(t)hd
s x +
G(χ(s))d
s x
(χ, ζ, h, G defined in Theorem 5.2). In fact, putting
H (t) ≡ E(t) +
γ +
1
2
< χ(t), χ(t) > +
1
2
< ψ 0 , ψ 0 > + < ω
−1 h, ω
−1 h >
(10.18)