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.