Blow-Up of Solution to Cauchy Problem for the Singularly Perturbed Sixth-Order Boussinesq-Type Equation ()
1. Introduction
In this paper, we consider the following Cauchy problem
(1.1)
(1.2)
where
is the unknown function,
is the given function,
and
are real numbers,
and
are given initial value functions.
In [1], the author has proved the existence and uniqueness of the global generalized solution and the global classical solution for the initial boundary value problem of Equation (1.1).
In [2], the author has discussed the nonexistence of global solution to the initial boundary value problem of Equation (1.1) in some condition.
In order to prove that blow-up of Cauchy problem (1.1), (1.2), we shall consider the following auxiliary problem
(1.3)
(1.4)
Then, we can obtain blow-up of the Cauchy problem (1.1), (1.2) from (1.3), (1.4) by setting
,
and
.
2. Main Theorems
Throughout this paper, we use the following notation:
. Now, we give the following main lemmas and theorems.
Lemma 2.1 (convex lemma [3]) Suppose that a positive twice-differential function
satisfies on
the inequality
(2.1)
where
and
are constants,
.
(1) If
and
, then there exist a
, such that
as
.
(2) If
and
, then
as
, where ![]()
and
![]()
Lemma 2.2 [4] Suppose that
, then
may be embedded into
, and for any
, we have
![]()
where
is a set of nonnegative integers.
Lemma 2.3 Suppose that
and
, then the solution
of the auxiliary problem (1.3), (1.4) satisfies the following energy identity
(2.2)
Proof Multiplying both sides of (1.3) by
, integrating on
, integrating by parts and lemma 2.2, we get
![]()
integrating the product over
, we get the identity (2.2).
Theorem 2.1 Suppose that
, and there exists
constant
and
, such that
(2.3)
Then, the solution
of the auxiliary problem (1.3), (1.4) blows-up in finite time if one of the following conditions holds
(1) ![]()
(2)![]()
(3) ![]()
Proof Suppose that the maximal time of the solution for (1.3), (1.4) is infinite. Let
(2.4)
where
and
are undetermined nonnegative constants. Differentiating (2.4) with respect to
, we have
(2.5)
By using the Hölder inequality, it follows from (2.5) that
(2.6)
Differentiating (2.5) with respect to
, making use of (1.3) and (2.2), we get
(2.7)
By virtue of interpolating inequality,
![]()
Observing the identity (2.7), we get
(2.8)
Combing (2.2), (2.3), (2.4), (2.6) with (2.8), we infer
(2.9)
(1) If
, by taking
, then
![]()
When
is sufficiently large,
. Clearly,
. It follows from lemma (2.1) that there exists
, such that
as
.
(2) If
, by taking
, we get
![]()
By virtue of assumption (2), we see
and
. It follows from lemma (2.1) that there exists
, such that
as
.
(3) If
, by taking
, (2.9) becomes
![]()
Defining
![]()
then
![]()
(2.10)
By virtue of assumption (3), we have
. Let
![]()
Thanks to the continuity of
,
is a positive number. Multiplying both sides of (2.10) by
, we find
(2.11)
Integrating (2.11) with respect to
over
, one gets
![]()
By virtue of assumption (3), we see that
![]()
Since
is a continuous function, we have for
,
(2.12)
It follows from the definition of
that (2.12) holds for all
. Integrating (2.12) with respect to
, we arrive at
![]()
Hence there is some
, such that
, where
![]()
So
becomes infinite at
.
Thus,
always becomes infinite at
under the assumption (1) or (2) or (3). This is a contradiction to the fact that the maximal time of existence of the solution is infinite. The theorem is proved.
Theorem 2.2 Suppose that
, and there exist constant
and
, such that
![]()
Then, the solution
of the Cauchy problem (1.1), (1.2) blows-up in finite time if one of the following conditions holds
(1) ![]()
(2)![]()
(3) ![]()
where
![]()
Proof Let
![]()
where
and
are nonnegative constants as those in Theorem 2.1.
By virtue of assumption Theorem 2.1,
satisfies the Equation (1.1) and the initial value condition (1.2) in classical sense. We take the change
(2.13)
Then
![]()
Substituting the above change (2.13) to the Cauchy problem (1.1), (1.2), we have
(2.14)
(2.15)
Integrating (2.14) and (2.15) over
, we obtain
(2.16)
(2.17)
Let
![]()
where
and
are nonnegative constants as those in Theorem 2.1. By virtue of assumption Theorem 2.1, the sufficient conditions of blow-up of solution to the Cauchy problem (2.16), (2.17) are fulfilled. Therefore, It follows from theorem 2.1 that
becomes infinite at
Since by the change (2.13),
, so
becomes infinite at
. Theorem 2.2 is proved.
Fund
This project is supported by NSF Grant 11271336, NSF of Henan Province Grant 122300410166.