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

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.


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  = 0, (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  = 1 in (1), attention is attracted to the wellknown 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 [7][8][9] and is completely integrable [10].It is regarded as a reexpression of geodesic flow on the diffeomorphism group of circle if  = 0 [11] and on the Bott-Virasoro group if  > 0 [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  0 ∈   (R) with  > 3/2 was given by several authors (see [13][14][15]).Under certain assumptions on initial data  0 , the equation has global strong solutions and blow-up solutions for periodic and nonperiodic case (see [13,[16][17][18][19][20][21][22]).The existence and uniqueness of global weak solutions in  1 (R) for the equation were proved (see [23][24][25]).It is shown from [6] that the solitary waves of the equation are peakon solitons and are orbitally stable.
If  = 0 and  ∈ R, (1) changes into the rod equation derived by Dai [26] recently, which describes finitelength 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.

Mathematical Problems in Engineering
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  = 3 and  = 0. 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 blowup solutions.More precisely, we establish the local wellposedness of strong solutions for (2) subject to initial value  0 ∈   (S),  > 3/2, with S = R/Z (the circle of unit length) and give a precise blow-up scenario.Under suitable assumptions on the initial value  0 , relying on the classical mathematical techniques, the several sufficient conditions about blow-up solutions are found.
Lemma 5. Let  ≥ 3/2 and (, ) is the corresponding solution of (4) with initial data  0 () ∈   (S); it holds that if  ∈ (0,  − 1], there is a constant  depending only on  such that Proof.We rewrite (2) in the following equivalent form: For  ∈ (0,  − 1], applying (Λ  )Λ  on both sides of (11) and integrating the new equation with respect to  by parts, we obtain 1 2 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  =  − 1, we have It follows from ( 19) and Gronwall's inequality that If there is a constant  > 0 such that ‖  ‖  ∞ ≤  on (0, ], then ‖‖ 2   (S) does not blow up.It completes the proof of Theorem 2.
Lemma 8 (see [2]). Moreover, Lemma 9. Let  ∈  3 (S) and  > 0 be the maximal existence time of the solution (, ) to problem (4).Then it holds that 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.
We now present the first blow-up result.
(i) If 0 <  < /3 and there is a  0 ∈ S such that then the corresponding solution to (2) blows up in finite time.
(ii) If /3 <  <  and there is a  0 ∈ S such that then the corresponding solution to (2) blows up in finite time.
(iii) If  < 0 or  >  and there is a  0 ∈ S such that then the corresponding solution to (2) blows up in finite time.
Next, we give the second blow-up result.
(i) If 0 <  < /3 and for all  > 0 there is a  0 ∈ S such that then the corresponding solution to (2) blows up in finite time.
(ii) If /3 <  <  and for all  > 0 there is a  0 ∈ S such that then the corresponding solution to (2) blows up in finite time.
(iii) If  < 0 or  >  and for all  > 0 there is a  0 ∈ S such that Next, we give the third blow-up result.
then the corresponding solution to (2) blows up in finite time.
then the corresponding solution to (2) blows up in finite time.
Proof.Let  > 0 be the maximal time of existence of the solution  to (2) with the initial data  0 .Applying  2    to both sides of (2) and integrating by parts, we get Since Therefore, we obtain Next, we divide (75) into three cases to prove the theorem.(i) The first is 0 <  ≤ /3; from case (i) of Theorem 11, we know that ( − 3) ≥ 0 and  * ((( − 3)/2) 2 ) ≥ 0.
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 (0) < −√2/3‖ 0 ‖  1  − , similar to the proof in case (i) of Theorem 11, we conclude that the corresponding solution will blow up in finite time.
where [x] stands for the integer part of  ∈ , then (1 −  2  ) −1  =  *  for all  ∈  2 (R) and  * ( −   ) = .Using this identity, (2) becomes 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.