In order to help the readers quickly get the main idea of the present paper, we show the main theorem in the beginning of this section.
Theorem 3.1 Let
, be given and let
. Assume that
, and the initial data satisfy
Then the existence time of a global solution for problem (1.1)-(1.2) is infinite.
In what follows, we show a preliminary lemma about the monotonicity of the functional , which will be used to prove the invariance of the new stable set
under the flow of problem (1.1)-(1.2).
Lemma 3.2 Let be given and be the solution of Equation (1.1) with initial data . Assume that and the initial data satisfy Equation (3.1), then the map is strictly decreasing as long as .
then we get
Note that by testing Equation (1
.1) with u
Then, Equation (3
. Obviously from
and (3.1) we can get
which implies . It is easy to see that , namely, . Therefore, we find that the map is strictly decreasing. □
Subsequently we show the invariance of the new stable set
under the flow of problem (1.1)-(1.2), which plays a key role in proving existence of global solutions for problem (1.1)-(1.2) at high initial energy level .
Lemma 3.3 (Invariant set)
Let be given and be a weak solution of problem (1.1)-(1.2) with maximal existence time interval , . Assume that the initial data satisfy (3.1). Then all solutions of problem (1.1)-(1.2) with belong to
, provided .
. If it is false, let
be the first time that
be defined as (3.2) above. Hence by Lemma 3.2, we see that
are strictly decreasing on the interval
. And then by (3.1), for all
, we have
Therefore from the continuity of
On the other hand, by (2.1) and (2.2) we can obtain
Recalling (3.6) we have
Then from the following equalities:
and (3.8), we can derive
. Or equivalently
which contradicts the first inequality of (3.7). This completes the proof. □
At this point we can prove the global existence for the solution of problem (1.1)-(1.2) with arbitrarily positive initial energy.
Proof of Theorem 3.1
From Theorem 2.1 there exists a unique local solution of problem (1.1)-(1.2) defined on a maximal time interval
be the weak solution of problem (1.1)-(1.2) with
and (3.1). Then from Lemma 3.3 we have
Therefore from (2.1)-(2.2) and (3.6), we can obtain
Hence from Theorem 2.1 it follows that and the solution of problem (1.1)-(1.2) exists globally. This completes the proof of Theorem 3.1. □