Conserved quantities, global existence and blow-up for a generalized CH equation

In this paper, we study conserved quantities, blow-up criterions, and global existence of solutions for a generalized CH equation. We investigate the classification of self-adjointness, conserved quantities for this equation from the viewpoint of Lie symmetry analysis. Then, based on these conserved quantities, blow-up criterions and global existence of solutions are presented.

1. Introduction. In this paper, we study the following shallow water equation with high-order nonlinearities, called a generalized Camassa Holm (g-CH) equation where b = 0 is a constant, m is a nonnegative integer, and u = u(t, x) is a function of t and x. Letting y := u − u xx sends equation (1) to the following g-CH family form y t + u m+1 y x + bu m u x y = 0, t > 0, x ∈ R.
It is a nonlinear dispersive wave equation that models the propagation of unidirectional irrotational shallow water waves over a flat bed [4,12,38,39], as well as water waves moving over an underlying shear flow [40]. The CH equation also arises in the In this paper, we study the conserved quantities, blow-up phenomena and global existence of solutions to equation (1). From the viewpoint of Lie symmetry and self-adjointness, we first construct some useful conservation laws of (1). Then by those conserved quantities we can analyze the global existence of solutions. In what follows, let us briefly present the notations, definition of nonlinear self-adjointness and Ibragimov's theorem on conservation laws. Consider a s-th order nonlinear equation E(x, u, u (1) , u (2) , · · · , u (s) ) = 0 (5) with n independent variables x = (x 1 , x 2 , · · · , x n ) and a dependent variable u = u(x), where u (s) = ∂ s u. A symmetry generator of (5) is denoted by Let E * (x, u, v, u (1) , v (1) , · · · , u (s) , v (s) ) := δL δu = 0 (7) be the adjoint equation of equation (5), where L = vE is called formal Lagrangian, v = v(x) is a new dependent variable and denotes the Euler-Lagrange operator.
Let us recall the conservation theorem given by Ibragimov in [34].
Theorem. (Ibragimov [34] ) Any Lie point, Lie-Bäcklund and non-local symmetry generated by of equation (5) provides a conservation law D i (C i ) = 0 for the system comprising equation (5) and its adjoint equation (7). The conserved vector C = (C i ) is given by where W = η − ξ j u j is the Lie characteristic function and L = vE is the formal Lagrangian.
We investigate the conserved quantities for equation (1) from the viewpoint of nonlinear adjointness, and give some classification of global existence of solution by the parameters b and m.
For the conserved quantities, we have the following classification result.
Especially, for m = 0 and b = 3, For b = m + 1, the related conservation laws are given by From the above theorem, we derive the following corollary.
Corollary 1. Let u be a solution of (1) and y = u − u xx , then it holds: for b = m + 1 or m = 0, b ∈ R, the mass for equation (1) is a constant; for m = 0 and b = 3, the weighted mass is conserved: and for b = m + 1, provided y m+1 b make sense.
Let us now consider the blow-up and existence results for equation (1). We first give a precise blow-up scenario.
This theorem covers blow-up scenario results to the CH equation in [9], DP equation in [56], b-equation in [20] and Novikov equation in [45]. It also similar to Theorem 1.2 in [59], but we give a different proof. It is very interesting that, for s > 5/2, we can show another precise blow-up scenario. Indeed, consider the associated Lagrangian scale of (1), and T > 0 is the maximal time of existence. We can derive a conserved quantity , which gives rise to a new precise blow-up scenario.
From Theorem 1.3 and 1.4, we easily obtain a global existence result.
Then the corresponding solution to equation (1) is defined globally in time.
For m = 0, our result generalizes partially the result on global existence for the b-equation (Theorem 4.3 in [20]). Moreover, we don't need the assumption Finally, based on the conserved quantities obtained in Corollary 1, we can present the following classifications of global existence results for solutions to equation (3), which also can be found in [59].
Then the solution of the problem (3) remains regular globally in time provided that one of the following conditions occurs: x )u 0 doesn't change sign on R; or (iv) b = m + 1; or b ∈ R and m = 0, additionally, y 0 doesn't change sign on R.
It is worth to note that Theorem 1.5 and (ii) in Theorem 1.6 don't cover each other.
The remainder of this paper is organized as follows. In Section 2, we discuss the nonlinear self-adjointness of equation (1) and find some conserved quantities based on this concept. In Section 3, we establish some results on blow-up scenario and global existence of solutions to equation (1) (or (3)).
2. Conserved quantities. Conserved quantities are very important and can be used to show some properties of solutions for some nonlinear equations, such as apriori estimates, global existence and stability of solutions. However, it seems not easy to find some useful conserved quantities for some nonlinear wave equations.
Here, we will apply the concepts of self-adjointness and the Ibragimov's theorem on conservation laws to construct some conserved quantities for equation (1).
First, we rewrite equation (1) directly and let v = v(t, x) be a new dependent variable. It's adjoint equation is given by where L = vE is the formal Lagrangian. Let λ i = λ i (t, x, u, u t , u x , · · · ) (i = 0, 1, 2, · · · ) be differential functions and, by the concept of nonlinear self-adjointness, we have Solving equation (27), we can see that: where C, C 1 and C 2 are arbitrary constants. Hence, we have demonstrated the following result.
Proof of Theorem 1.1. Theorem 1.1 is deduced immediately from above proposition.
Now we investigate the symmetries of equation (1) by the classical Lie symmetry analysis [48]. Consider the vector field which has the fourth-order prolongation, from (1), x , η xx , η xxt and η xxx can be expressed via the components of the vector field ξ t , ξ x and η. The invariant condition leads to the following classification result.
Next, from equation (3) we consider the following system Assume that the form Lagrangian of this system is L s = wE 1 + zE 2 , where w = w(t, x) and z = z(t, x) are two dependent variables. Then the adjoint system of (29) is By the definition of nonlinear self-adjointness, we have where h = h(t, x, u, u t , u x , · · · ), g = g(t, x, u, u t , u x , · · · ), µ i and τ i (i = 1, 2) are differentiable functions.
On the other hand, let be the symmetry operator of (29). From system (31), we get Solving this system and going through the process of classical Lie symmetry analysis, we can obtain the following result. , z = 0. Moreover, the symmetries admitted by (29) are given by where f (t) is a differentiable function.

LONG WEI, ZHIJUN QIAO, YANG WANG AND SHOUMING ZHOU
Based on Propositions 1-3, we now can apply Ibragimov's theorem to construct some conservation laws for equation (1).
Proof of Theorem 1.2. Rewriting the formal Lagrangian L = vE in the symmetric form where v is given in Proposition 1. From Ibragimov's theorem, for a general generator X given by (28), we have that the Lie characteristic function W = η − ξ t u t − ξ x u x , the density is given by and the flux is For the symmetries X 1 -X 5 , from the formula (34) and (35) we obtain readily some conservation laws for equation (1).
For the case m = 0, b ∈ R or b = m + 1, the substitution function is v = 1 (we take C = 1). Let us construct the conserved vector corresponding to the time translation group with the generator X 1 = ∂ t . For this operator, we have W = −u t . Therefore, we obtain the following conserved vector which can be simplified to (note that m = 0, b ∈ R or b = m + 1 in this case) C t = 0, C x = 0, that is, this conservation law is trivial. In this case, for the generator X 2 = ∂ x , we also show that the corresponding conservation law is trivial. However, for X 3 = −(m + 1)t∂ t , from the formula (34) and (35) we get the conservation law as follows which can be reduced to , which is given by (13). The other conservation laws in the theorem can be given by similar processes to above. As for (C t 5 , C x 5 ), we need to use Proposition 3. By the Theorem of Ibragimov, we obtain that (3) it is easy to check that

Moveover, in view of equation
makes sense. In another ward, they are also conservation laws for equation (3).
Proof of Corollary 1. From Theorem 1.2, we can derive some conserved integrations. For b = m + 2, integrating D t C t + D x C x = 0 over R and applying the conditions for u at infinity, it's easy to find the H 1 norm of u is conserved, that is, Similarly, from C t 2 , we can see that which means that the weighted This shows that the mass is conserved for equation (1). From C t 4 , we see that the weighted mass is conserved, R e ±2x udx = C 4 , for m = 0 and b = 3.
Finally, we deduce from C t 5 , C t 5,1 and C t 5,2 that for b = m + 1 3. Blow-up scenario and global existence. In this section we derive the precise blow-up scenario and show some global existence results for the strong solutions to equation (3). Denote the operator (1 − ∂ 2 x ) 1 2 by Λ and the kernel of Λ −2 by G := 1 2 e −|x| . Then Λ −2 f = G * f for all f ∈ L 2 (R) and G * y = u. Using this identity, equation (3) can be reformulated in the following form or in the equivalent form where First, we recall the local well-posedness of the Cauchy problem for equation (3) in H s (R), s > 3 2 . By Kato's semigroup theorem or Galerkin-type approximation scheme, the authors of [26,57] showed the following result.
We will need the following useful lemmas.
where c is a constant depending only on s.
where where [A, B] denote the commutator of linear operator A and B, c is a constant depending only on s. Now, let us consider the associated Lagrangian scale of (1), that is, the initial value problem ∂q(t, x) ∂t = u m+1 (t, q(t, x)), x ∈ R, 0 < t < T, where u ∈ C([0, T ); H s ) is the solution of (1) with initial data u 0 ∈ H s with s > 3 2 and T > 0 is the maximal time of existence. A direct calculation shows that q tx (t, x) = (m + 1)u m u q (t, q(t, x))q x (t, x). So, for t ∈ [0, T ), x ∈ R, we have q x (t, x) = e (m+1) t 0 u m uq(τ,q(τ,x))dτ > 0.
In view of equations (3) and (41), one gets easily that d dt [y(t, q(t, x))q x (t, x) b m+1 ] = (y t + u m+1 y x + bu m u x y)q b m+1 x = 0, which means that We now consider some global existence results for the strong solutions to equation (3). Let us state a global existence result.
then the H s -norm of u(·, t) does not blow up on [0, T ).
Proof. The idea of the proof is classical and is similar to those used in [20,59] and the references therein. However, there seems to be a gap in the proof of Theorem 4.1 in [59]. So we give the details of the proof here. Let u be the solution to equation (39) with the initial data u 0 ∈ H s , s > 3 2 . And let T be the maximal existence time of the corresponding solution u, which is guaranteed by Theorem 3.1. Assume that (44) holds. Applying the operator Λ s to equation (39), multiplying by Λ s u, and integrating over R yield that where and g, h are given by (40). For the first term of right side in (45), from Lemma 3.3 we have the following estimate: which can be obtained by applying Lemma 3.2 repetitiously. As for the second term on the right in (45), when m = 0 it would be disappear; when m = 1, it can be estimated, we refer to the proof of Theorem 3.1 in [53]. Now, we assume here that m ≥ 2. It follows from the Hölder inequality that Thanks to Lemma 3.2, we see that here we have used estimate which has been obtained similarly to (46). Thus, we obtain (45), we need to estimate two parts |(∂ x Λ −2 u m+2 , u) s | and |(∂ x Λ −2 u m u 2

For the last term in
x , u) s | respectively. First, from Lemma 3.3 we see that We have also used the estimate similar to (46) here. We next estimate the term |(∂ x Λ −2 (u m u 2 x ), u) s |. Note that H s and H s−1 are algebraic with s > 3 2 . So we have . Therefore, we see that from above estimates Then from the Gronwall's inequality and assumption of the theorem, we obtain provided (44) holds. This completes the proof of the theorem. Now, we prove Theorem 1.3-1.6.
Proof of Theorem 1.3. By Theorem 3.4 and a simple density argument, it is needed only to show the desired result is valid when s ≥ 2. Note y = u − u xx , it's easy to see that For b = m+1 2 , taking account for (43) we know that y 2 L 2 = R y 2 (t, q(t, s))q s (t, s)ds = R y 2 0 (s)q s (t, s) 1− 2b m+1 ds.
Due to the equality (42), one has y L 2 = y 0 (x)e ( m+1 2 −b) t 0 (u m uq)(τ,q(τ,x))dτ Assume (23) is not valid. Then there is some positive number M 1 > 0 such that which combining Theorem 3.4 shows that the solution does not blow up in finite time for s ≥ 2.
Conversely, the Sobolev embedding theorem H s (R) → L ∞ (R) (with s > 1 2 ) implies that if (23) holds, the corresponding solution blows up in finite time, which completes the proof.
Proof of Theorem 1.4. Let u be the solution to equation (39) with the initial data u 0 ∈ H s , s > 5 2 and T be the maximal existence time of the corresponding solution u.
Thanks to the equalities (42) and (43), we have 0 (u m uq)(τ,q(τ,x))dτ Suppose that (24) is not valid. Then there exists a positive number M 2 > 0 such that Note that u = G * y and u x = ∂ x G * y. From the Young inequality, it is easy to see that u L ∞ ≤ G L 1 y L ∞ = y L ∞ and u x L ∞ ≤ ∂ x G L 1 y L ∞ = y L ∞ , which combining (52) imply On the other hand, since u 0 ∈ H s and s > 5/2, by the Sobolev embedding theorem we have This together with (53) and Theorem 3.4 completes the proof.
Proof of Theorem 1.5. The result is obtained immediately from Theorem 1.3 and 1.4.
Proof of Theorem 1.6. Applying the conserved quantities in Corollary 1 and some analysis technique, one easily obtains the global existence results, for the details we refer to [59].