Wave-Breaking Phenomena and Existence of Peakons for a Generalized Compressible Elastic-Rod Equation

and Applied Analysis 3 The following blow-up criterion shows that the wavebreaking depends only on the infimum of γu x . Theorem 7. Let u 0 ∈ H s (R) be as in Theorem 3 with s > 3/2. Then the corresponding solution u to (5) blows up in finite time T ∗ u0 > 0 if and only if lim inf t↑T ∗ u0 inf x∈R {γu x (t, x)} = −∞. (9) Proof. Since, in view of Remark 5, the existence time T∗ u0 is independent of the choice of s, we need only to consider the case s = 3, which relies on a simple density argument. Multiplying (3) bym and integrating over R with respect to x and then integration by parts produce


Introduction
Consideration herein is the following generalized hyperelastic-rod wave equation: where  : R → R is a given  ∞ -function and the real number  with  ̸ = 0 is a given parameter.In general, the constant  is given in terms of the material constants and the prestress of the rod.In fact, the authors [1,2] studied the special compressible materials which lead to values of  ranging from −29.4760 to 3.4174.
The classical CH equation (2) was originally proposed as a model for surface waves [3] and has been studied extensively in the last twenty years because of its many remarkable properties: infinity of conservation laws and complete integrability [3], existence of peaked solitons and multipeakons [3,5,6] (with  = 0), well-posedness and breaking waves, and meaning solutions that remain bounded, while its slope becomes unbounded in finite time [7,8].On the other hand, if  = 0, the CH equation (2) admits peaked solitary wave solutions (called peakons) which possess the form (, ) =  −|−| with speed  ∈ R,  ̸ = 0, and their stability was studied in [9].Recently, Gui et al. [10] proved that there exist some peaked functions which are global weak solutions to a modified Camassa-Holm equation.We should mention that the solutions to the CH equation ( 2) can be uniquely continued after wave-breaking as either a global conservative or global dissipative weak solution [11][12][13][14].It is worth pointing out that there exists a global-in-time weak solution to the CH equation in the energy space [15].
For  ∈ R\{0}, () = 2+(3/2) 2 , (1) serves as a model equation for mechanical vibrations in a hyperelastic rod [1,2].Similar to CH equation, stability of solitary wave solutions has been studied in [16].In addition, the solutions to the hyperelastic-rod wave equation can be uniquely continued after wave breaking as a global conservative weak solution 2 Abstract and Applied Analysis [17].Moreover, there exists a global-in-time weak solution to the hyperelastic-rod wave equation in the energy space [18].
Motivated by the approaches in [9,10], our goals in this paper are concerned with the wave-breaking phenomena and the existence of some new peakons of (1) with () =  2 + 2 (,  ∈ R,  > 0),  ̸ = 0.In this case, (1) may be read as Introducing the momentum  :=  −   , we get from (3) that Note that if () := (1/2) −|| ,  ∈ R, then (1− 2  ) −1  =  *  for all  ∈  2 (R) and  * ( −   ) = , where * denotes convolution with respect to the spatial variable .Therefore, (3) can also be rewritten as the following equivalent form We are now in a position to give the notions of strong and weak solutions.
) is said to be a weak solution to the initial-value problem (5) if it satisfies the following identity: for any smooth test function (, ) ∈  ∞  ([0, ) × R).If  is a weak solution on [0, ) for every  > 0, then it is called a global weak solution.
Our main results of the present paper are Theorem 9 (wave breaking) and Theorem 10 (existence of peakons).
The remainder of the paper is organized as follows.In Section 2, the results of blow-up to strong solutions are presented in detail.It is shown that the solutions of (3) can only have singularities which correspond to wave breaking (Theorems 9).In Section 3, the existence of some new peaked solutions of (3) is verified.From this, we know that there exist some peaked solitary wave solutions for the case  ̸ = 0 (compared to the case in the Camassa-Holm equation; see Remark 11).
Notation 1.As above and henceforth, we denote the norm of the Lebesgue space   (R) (1 ≤  ≤ ∞) by ‖ ⋅ ‖   and the norm of the Sobolev space   (R) ( ∈ R) by ‖ ⋅ ‖   .We denote by * the spatial convolution on R.

Wave-Breaking Phenomena
By using the Kato's method [19], we may easily get the following results about the local well-posedness and blowup criterion of strong solutions to (5), of which proofs are similar to the one as in the CH equation in [7] (up to a slight modification) and we omit it here.
Then there exists a time  > 0 such that the initial-value problem (5) We are now in a position to state a blow-up criterion for (5).Theorem 4. Let  0 ∈   (R) be as in Theorem 3 with  > 3/2.Let  be the corresponding solution to (5).Assume that  *  0 > 0 is the maximum time of existence.Then Remark 5.The blow-up criterion (7) implies that the lifespan  *  0 does not depend on the regularity index  of the initial data  0 .Indeed, let  0 be in   for some  > 3/2 and consider some   ∈ (3/2, ).Denote by   (resp.,    ) the corresponding maximal   (resp.,    ) solution given by the above theorem.
Abstract and Applied Analysis 3 The following blow-up criterion shows that the wavebreaking depends only on the infimum of   .Theorem 7. Let  0 ∈   (R) be as in Theorem 3 with  > 3/2.Then the corresponding solution  to (5) blows up in finite time  *  0 > 0 if and only if Proof.Since, in view of Remark 5, the existence time  *  0 is independent of the choice of , we need only to consider the case  = 3, which relies on a simple density argument.
Multiplying (3) by  and integrating over R with respect to  and then integration by parts produce We next differentiate (3) with respect to  to get Multiplying by   then integrating over R with respect to  lead to Therefore, If   is bounded from below on [0,  *  0 ) × R, that is, there exists a positive constant where we used (8) for  ∈ [0,  *  0 ), which ensures that the solution (, ) does not blow up in finite time.
On the other hand, if lim inf by Theorem 3 for the existence of local strong solutions and the Sobolev embedding theorem, we infer that the solution will blow up in finite time.The proof of Theorem 7 is then complete.
For  ̸ = 0, we define where sign() is the sign function of  ∈ R, and set s 0 := s(0).Then, thanks to Theorem 3, for every  ∈ [0, ) there exists at least one point () ∈ R with s() =   (, ()).Just as the proof given in [8], one can show the following property of s().
Indeed, observe that the inequality Whereas the inequality which along with (34) gives rise to (32).Notice that Repeating the above argument in the proof of (31) for the case  ≥  leads to (31), and then the proof of Theorem 9 is complete for the case 0 <  < .
Let us now consider the case  < 0.
Repeating the above argument in the proof of (31) for the case  ≥  again ends the proof of Theorem 9.

Peaked Solitary Wave Solution
In this section we consider the existence of peaked solitary wave solutions of (3) in the case of () =  2 +2 (,  ∈ R and  ̸ = 0), which can be understood as global weak solutions.Proof of Theorem 10.The proof of the theorem is motivated by the method in [10].First, we can reduce the result to the