For the equation in the title, let P and A be positive semidefinite operators (with P strictly positive) defined on a dense subdomain
D
⊆
H
D \subseteq H
, a Hilbert space. Let D be equipped with a Hilbert space norm and let the imbedding be continuous. Let
F
:
D
→
H
\mathcal {F}:D \to H
be a continuously differentiable gradient operator with associated potential function
G
\mathcal {G}
. Assume that
(
x
,
F
(
x
)
)
≥
2
(
2
α
+
1
)
G
(
x
)
(x,\mathcal {F}(x)) \geq 2(2\alpha + 1)\mathcal {G}(x)
for all
x
∈
D
x \in D
and some
α
>
0
\alpha > 0
. Let
E
(
0
)
=
1
2
[
(
u
0
,
A
u
0
)
+
(
v
0
,
P
v
0
)
]
E(0) = \tfrac {1}{2}[({u_0},A{u_0}) + ({v_0},P{v_0})]
where
u
0
=
u
(
0
)
,
v
0
=
u
t
(
0
)
{u_0} = u(0),{v_0} = {u_t}(0)
and
u
:
[
0
,
T
)
→
D
u:[0,T) \to D
be a solution to the equation in the title. The following statements hold: If
G
(
u
0
)
>
E
(
0
)
\mathcal {G}({u_0}) > E(0)
, then
lim
t
→
T
−
(
u
,
P
u
)
=
+
∞
{\lim _{t \to {T^ - }}}(u,Pu) = + \infty
for some
T
>
∞
T > \infty
. If
(
u
0
,
P
v
0
)
>
0
,
0
>
E
(
0
)
−
G
(
u
0
)
>
α
(
u
0
,
P
v
0
)
2
/
4
(
2
α
+
1
)
(
u
0
,
P
u
0
)
({u_0},P{v_0}) > 0,0 > E(0) - \mathcal {G}({u_0}) > \alpha {({u_0},P{v_0})^2}/4(2\alpha + 1)({u_0},P{u_0})
and if u exists on
[
0
,
∞
)
[0,\infty )
, then (u,Pu) grows at least exponentially. If
(
u
0
,
P
v
0
)
>
0
({u_0},P{v_0}) > 0
and
α
(
u
0
,
P
v
0
)
2
/
4
(
2
α
+
1
)
(
u
0
,
P
u
0
)
≤
E
(
0
)
−
G
(
u
0
)
>
1
2
(
u
0
,
P
v
0
)
2
/
(
u
0
,
P
u
0
)
\alpha {({u_0},P{v_0})^2}/4(2\alpha + 1)({u_0},P{u_0}) \leq E(0) - \mathcal {G}({u_0}) > \tfrac {1}{2}{({u_0},P{v_0})^2}/({u_0},P{u_0})
and if the solution exists on
[
0
,
∞
)
[0,\infty )
, then (u,Pu) grows at least as fast as
t
2
{t^2}
. A number of examples are given.