Constructions of harmonic maps between Hadamard manifolds

1.2. GLOBAL PROPERTIES OF SOLUTIONS 33
for any nonnegative integers m and n. As a consequence, we have
{f 1 (t) p f 2 (t) q ṙ(t)} 2 ≤{f 1 (t 1 ) p f 2 (t 1 ) q ṙ(t 1 )} 2
Letting t 1 tend to 0, we have
+ µ 2 {f 1 (t) 2p−2 f 2 (t) 2q h 1 (r(t)) 2 − f 1 (t 1 ) 2p−2 f 2 (t 1 ) 2q h 1 (r(t 1 )) 2 }
+ ν 2 {f 1 (t) 2p f 2 (t) 2q−2 h 2 (r(t)) 2 − f 1 (t 1 ) 2p f 2 (t 1 ) 2q−2 h 2 (r(t 1 )) 2 }.
{f 1 (t) p f 2 (t) q ṙ(t)} 2 ≤ µ 2 f 1 (t) 2p−2 f 2 (t) 2q h 1 (r(t)) 2 + ν 2 f 1 (t) 2p f 2 (t) 2q−2 h 2 (r(t)) 2 .
Dividing both sides by f 1 (t) 2p f 2 (t) 2q , we obtain the conclusion.
Using this lemma, we present a sufficient condition for the existence of a global solution
in terms of a relation between t 0 and r 0 .
If r(t) = 0 for some t>0, then r(t) ≡ 0 by Corollary 1.1.3. Hence it is a global solution.
In the sequel of this section we assume that r(t) > 0 for all t>0.
Let
⎧
⎪⎨ max{h 1 (r),h 2 (r)} (ν >0),
h(r) =
⎪⎩ h 1 (r) (ν =0).
Since ṙ(t) > 0, we have
{
ṙ(t) 1
h(r(t)) ≤ γ f 1 (t) + 1 }
,
f 2 (t)
where γ = max{µ, ν}. Integrating both sides from t 0 to t(∈ [t 0 ,T]), we obtain
∫ r(t) ∫
ds
t
{ 1
(1.2.3)
h(s) ≤ γ t 0
f 1 (τ) + 1 }
dτ.
f 2 (τ)
r 0
Theorem 1.2.3. If r 0 satisfies
∫ ∞ ∫
dr
∞
{ 1
(1.2.4)
r 0
h(r) >γ t 0
f 1 (τ) + 1 }
dτ,
f 2 (τ)
then the solution to the equation (1.1.1) with the boundary condition (1.1.2) exists globally
and is bounded.