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Mathematical Problems in Engineering
Volume 2014, Article ID 751949, 11 pages
http://dx.doi.org/10.1155/2014/751949
Research Article

On Blow-Up Structures for a Generalized Periodic Nonlinearly Dispersive Wave Equation

1Department of Applied Mathematics, Sichuan University of Science and Engineering, Zigong 643000, China
2Sichuan Province University Key Laboratory of Bridge Non-Destruction Detecting and Engineering Computing, Zigong 643000, China

Received 6 November 2013; Accepted 23 February 2014; Published 7 April 2014

Academic Editor: Shaoyong Lai

Copyright © 2014 Ying Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The local well-posedness for a generalized periodic nonlinearly dispersive wave equation is established. Under suitable assumptions on initial value , a precise blow-up scenario and several sufficient conditions about blow-up results to the equation are presented.

1. Introduction

Recently, Hu and Yin [1] and Yin [2] investigate the following equation: where is nonnegative number and is arbitrary real number. It is shown from [1, 2] that (1) has solitary wave solutions and blow-up solutions for nonperiodic case and also solutions which blow up in finite time for periodic case.

If , (1) becomes the famous BBM equation modelling the motion of internal gravity waves in shallow channel [3]. Some results related to the equation can be found in [4, 5]. It is worthwhile to mention that the equation does not have integrability and its solitary waves are not solitons [5].

If in (1), attention is attracted to the well-known Camassa-Holm equation, which models the unidirectional propagation of shallow water waves over a flat bottom. Here, represents the free surface above a flat bottom and is a nonnegative parameter related to the critical shallow water speed [6]. As a model to describe the shallow water motion, the Camassa-Holm equation possesses a bi-Hamiltonian structure and infinite conservation laws [79] and is completely integrable [10]. It is regarded as a reexpression of geodesic flow on the diffeomorphism group of circle if [11] and on the Bott-Virasoro group if [12]. Recently, some significant results of dynamical behaviors have been obtained for the Cauchy problem of the Camassa-Holm equation. For example, the local well-posedness of corresponding solution for initial data with was given by several authors (see [1315]). Under certain assumptions on initial data , the equation has global strong solutions and blow-up solutions for periodic and nonperiodic case (see [13, 1622]). The existence and uniqueness of global weak solutions in for the equation were proved (see [2325]). It is shown from [6] that the solitary waves of the equation are peakon solitons and are orbitally stable.

If and , (1) changes into the rod equation derived by Dai [26] recently, which describes finite-length and small amplitude radial deformation waves in thin cylindrical compressible hyperelastic rods (see [26]), and represents the radial stretch relative to a prestressed state in one-dimensional variable. The first investigation of the Cauchy problem of the rod equation on the line was done by Constantin and Strauss [27], the precise blow-up scenario, some blow-up results of strong solution, and the stability of a class of solitary waves to the rod equation are presented. In [28, 29], Zhou found the sufficient conditions to guarantee the finite blow-up of corresponding solution for periodic case. Moreover, Yin [30] discusses the rod equation on the circle and gives some interesting blow-up results.

In this paper, we consider a generalized nonlinearly dispersive wave equation on the circle where and are nonnegative fixed constants and and are fixed arbitrary constants. Obviously, (2) reduces to (1) if we define and . Actually, Wu and Yin [31] consider a nonlinearly dissipative Camassa-Holm equation which includes a nonlinearly dissipative term , where is a differential operator. Thus, we can regard the term as a dissipative term.

Because of the term , (2) does not admit conservation laws in previous works [1, 2]: Several estimates are established to prove several blow-up solutions. More precisely, we establish the local well-posedness of strong solutions for (2) subject to initial value , with (the circle of unit length) and give a precise blow-up scenario. Under suitable assumptions on the initial value , relying on the classical mathematical techniques, the several sufficient conditions about blow-up solutions are found.

2. Local Well-Posedness

In this section, we establish the local well-posedness and blow-up scenario for the Cauchy problem (2) in , .

We denote by the convolution. Note that if , where [] stands for the integer part of , then for all and . Using this identity, (2) becomes which is equivalent to

Theorem 1. Given , there exist a maximal and a unique solution to problem (2), such that

Proof. The proof of Theorem 1 can be finished by using Kato's semigroup theory (see [1] or [2]). Here, we omit the detailed proof.

3. Blow-Up Solutions

Theorem 2. Let ; the solution of of problem (2) is uniformly bounded. Blow-up in finite time occurs if and only if

Before proving the theorem, we give several useful lemmas.

Lemma 3 (Kato and Ponce [32]). If , then is an algebra. Moreover where is a constant depending only on .

Lemma 4 (Kato and Ponce [32]). Let . If and , then

Lemma 5. Let and is the corresponding solution of (4) with initial data ; it holds that if , there is a constant depending only on such that

Proof. We rewrite (2) in the following equivalent form: For , applying on both sides of (11) and integrating the new equation with respect to by parts, we obtain where the Parseval equality is used. We will estimate each of the terms on the right-hand side of (11). For the first and the third terms, using integration by parts, the Cauchy-Schwartz inequality, and Lemmas 3 and 4, we have where only depends on . Using the above estimate to the second term yields For the fourth term, using Lemma 3 gives rise to It follows from (12)–(16) that which leads to It completes the proof of the lemma.

Proof of Theorem 2. Applying (10) with , we have It follows from (19) and Gronwall’s inequality that If there is a constant such that on , then does not blow up. It completes the proof of Theorem 2.

Lemma 6 (see [17]). Let be the solution to (2) on with initial data , . Then there exists at least one point with The function is almost everywhere differentiable on with

Lemma 7 (see [21, 22]). (i) For every , one has where the constant is sharp.
(ii) For every , one has with the best possible constant lying within the range . Moreover, the best constant is .

Lemma 8 (see [2]). If is such that , then, for every , one has Moreover,

Lemma 9. Let and be the maximal existence time of the solution to problem (4). Then it holds that (i), (ii).

Proof. The proof of (i) is similar to that of [2, lemma 3.6], so we omit it.
Multiplying to both sides of (2) and integrating by parts, we get which yields It finishes the proof.

Lemma 10 10 (see [29]). Assume that a differential function satisfies with constants . If the initial datum , then the solutions to (29) go to in finite time.

We now present the first blow-up result.

Theorem 11. Let and , .(i)If and there is a such that then the corresponding solution to (2) blows up in finite time.(ii)If and there is a such that then the corresponding solution to (2) blows up in finite time.(iii)If or and there is a such that then the corresponding solution to (2) blows up in finite time.

Proof. Let be the maximal time of existence of the solution to (2) with the initial data . Note that . Differentiating (4) with respect to and multiplying the obtained equation by , we get Noting that , for all , and defining , we obtain Next, we divide (34) into three cases to prove the theorem.
(i)The first is  ; note that   and . Then, we have Note that . Thus, we obtain From Lemmas 8 and 9, we have Thus, Note that . Using Young’s inequality, we get
Therefore, we deduce which results in with Note that, from Lemma 10, if , then there exists , such that . Applying Theorem 2, the solution to (2) does not exist globally in time.
(ii) The second is  . Similar to case (i), we have and
Therefore, we obtain
Using (37)–(39), we get which results in with Note that, from Lemma 10, if , then there exists , such that . Repeating the proof of (i), we conclude that the solution blows up in finite time.
(iii) The third is   or ; note that . From Young’s inequality, we get for all Using (37)–(39), we get which results in with Note that, from Lemma 10, if , then there exists , such that . Repeating the proof of (i), we deduce that the solution blows up in finite time. It finishes the proof of the theorem.

Next, we give the second blow-up result.

Theorem 12. Let and , , and .(i)If and for all there is a such that then the corresponding solution to (2) blows up in finite time.(ii)If and for all there is a such that then the corresponding solution to (2) blows up in finite time. (iii)If or and for all there is a such that then the corresponding solution to (2) blows up in finite time.

Proof. Similar to Theorem 11, we divide (34) into three cases to prove the theorem.
(i) The first is  ; note that and . Then, we have .
From Lemmas 8 and 9, we have Thus, Note that . Using Young’s inequality, we get
Therefore, we deduce which results in with Note that, from the lemma, if , then there exists , such that . Applying Theorem 2, the solution blows up in finite time.
(ii) The second is  . Similar to case (i), we have and
Using (55)–(57), we get which results in with Following the proof of (i) in Theorem 11, we derive that the solution blows up in finite time.
(iii) The third is   or ; note that . From Young’s inequality, we get for all Using (55)–(57) and (65), we get which results in with Following the proof of (i) in Theorem 11, we obtain that the solution blows up in finite time. It finishes the proof of the theorem.

Next, we give the third blow-up result.

Theorem 13. Assume that and , , and .(i)If is such that then the corresponding solution to (2) blows up in finite time.(ii)If is such that then the corresponding solution to (2) blows up in finite time.(iii)If or is such that then the corresponding solution to (2) blows up in finite time.

Proof. Let be the maximal time of existence of the solution to (2) with the initial data . Applying to both sides of (2) and integrating by parts, we get Since thus
Therefore, we obtain Next, we divide (75) into three cases to prove the theorem.
(i) The first is   ; from case (i) of Theorem 11, we know that and .
From Holder's inequality and Yong's inequality, we have
Using (76) and (75), it yields
Setting we have
Note that, from the lemma, if , similar to the proof in case (i) of Theorem 11, we conclude that the corresponding solution will blow up in finite time.
(ii) The second is  ; from case (ii) of the theorem, we have and .
Using (76) and (75), it yields
Setting we have
Note that, from the lemma, if , similar to the proof in case (i) of Theorem 11, we derive that the corresponding solution will blow up in finite time.
(iii) The third is   or ; note that .
Using (76) and (75), it yields
Setting we have
Note that, from the lemma, if , similar to the proof in case (i) of Theorem 11, we deduce that the corresponding solution will blow up in finite time. It completes the proof of Theorem 13.

Finally, we give the fourth blow-up result.

Theorem 14. Let and , . Assume that and . If and satisfy one of the following conditions:(i) and ,(ii) and ,(iii) and ,then the corresponding solution of (2) blows up in finite time.

Proof. Supposing that the statement is not correct, then the solution of (2) exists globally in time.
Thus, applying to both sides of (4) and integrating by parts, we get
Due to and (see [29]), we get Since , we have Therefore, we deduce Next, we divide (90) three cases to prove the theorem.
(i) The first is   . Note that and . From the condition of Theorem 14, we have where . From Holder's inequality, we obtain Thus, we have From (93), we have Solving (94), we obtain where is constant which implies that there is a such that On the other hand, Solving (97), we obtain where is constant, which leads to a contradiction as becomes large enough. Therefore, case (i) is proved.
(ii) The second is  . Note that and . From the condition of the theorem, we have where . Repeating the arguments in the proof of case (i), we deduce that the corresponding solution blows up.
(iii) The third is  . Note that and . From the condition of the theorem, we have