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Wave breaking for shallow water models with time decaying solutions

A family of Camassa-Holm type equations with a linear term and cubic and quartic nonlinearities is considered. Local well-posedness results are established via Kato's approach. Conserved quantities for the equation are determined and from them we prove that the energy functional of the solutions is a time-dependent, monotonically decreasing function of time, and bounded from above by the Sobolev norm of the initial data under some conditions. The existence of wave breaking phenomenon is investigated and necessary conditions for its existence are obtained. In our framework the wave breaking is guaranteed, among other conditions, when the coefficient of the linear term is sufficiently small, which allows us to interpret the equation as a linear perturbation of some recent Camassa-Holm type equations considered in the literature.


Introduction
Indubitably the Camassa-Holm (CH) equation is one of the cutting edge topics in mathematics and mathematical physics in general, and differential equations and applied analysis in particular.
The work of Camassa and Holm [4], where the equation was derived as a model for shallow water waves, can be considered as of fundamental nature and is the cornerstone of an active field of investigation of non-local integrable equations. The discovery of the CH equation was the prelude of a myriad of other non-local integrable equations, such as the Degasperis-Procesi (DP) [16,17] and the Novikov equations [24,33]. They share the following properties with the CH equation: bi-Hamiltonian structure, integrability, existence of peakon and multipeakon solutions, and infinite hierarchy of conservation laws, to name a few.
The CH equation was firstly deduced by Fuchsteiner and Fokas in [20]. It was, in fact, a formal deduction, without physical motivation. However, on the grounds of physics, in [4] the equation was recovered as a unidirectional model for describing the height of water's free surface above the flat bottom for a shallow water system, see also [26].
One of the interesting properties of the CH equation, and other similar equations not necessarily among the ones mentioned before, is the fact that they have many conservation laws. This is very useful and vital to establish qualitative results about the solutions of equations and partially explains why this sort of equations are particularly fashion in the field of differential equations.
We would like to recall some simple, but extremely important concepts in differential equations where time is involved. Given a differential equation and an initial data (that is, a Cauchy problem), some questions are of capital importance: 1. Does the problem have a solution? This is the question of existence.
2. If there is a solution, is it unique? This is the problem of uniqueness.
3. If there is a solution, does it depend continuously on the initial data?
These three essential questions are central when one is investigating qualitative behaviour of the solutions of equations subject to the restrictions imposed by a certain condition. In this paper, the equation we shall study depends on t and x and the condition we shall consider is an initial condition, that is, we know how the solution behaves at t = 0. In case these questions are satisfied, the problem under consideration is said to be well-posed (in the sense of Hadamard).
Although the questions above are basic and fundamental, they are not necessarily simple or easy to be addressed. In particular, very often the solution u of a problem like the one we shall deal with in this paper belongs to C 0 ([0, T ), E), meaning that u(·, x) ∈ C 0 ([0, T )) and u(t, ·) ∈ E, where E is a usually a Banach space, whereas T > 0 is the maximal time of existence of the solution, or lifespan, which usually depends on the initial data and the space E.
The value of the lifespan add another ingredient to our menu: If the problem is well-posed, what happens in case T < ∞? What if T = ∞? In the first case we have a local well-posed problem: it is well-posed provided that t ∈ [0, T ), while in the second we have a global well-posed problem, meaning that the solution exists for any t.
In case we only have local well-posedness, we say that the solution u of the problem develops a finite time blow-up. This happens just because the solution cannot be described for all values of t.
The blow-up phenomena can have different manifestations, depending on the problem. It can arise, for example, if the solution of the problem becomes unbounded when t approximates T . More specifically, a blow-up in finite time turns out if T < ∞ and lim sup t→T u(t, ·) E = ∞.
A blow-up can also arise in the following situation: Assume that the Banach space E is a subspace of C 1 (R) and T < ∞. Then we have another sort of blow-up if This kind of blow-up, which takes the x−derivative of the solution into account, is best known as wave breaking. From a geometrical viewpoint, it means that the tangent line to the curve x → (x, u(t, x)) tends to the perpendicular line to the x−direction when t approaches T .

Notation and conventions
Throughout this paper, if u is a function depending on two variables, both u t and ∂ t u denote the derivative of u with respect to its first argument, while u x and ∂ x u mean the derivative with respect to the second independent variable.
The norm in a Banach space E is denoted by · E , whereas ·, · H means the inner product in a Hilbert space H. If E and F are two Banach spaces, the set of bounded linear operators from E into F is denoted by L(E, F ), or L(E) in the case F = E.
The space of test functions ϕ : R → R is denoted by S(R), whereas its dual topological space is refereed as S ′ (R). A member of S(R) is known as rapidly decreasing smooth function, while members of S ′ (R) are called tempered distributions.
Making use of the auxiliary variable m = u − u xx we reformulate the object of investigation of this paper as the following: We aim at investigating properties of the initial value problem, Our first aspiration is to address the three questions listed in the beginning of the paper. We prove that (1.1) is locally well-posed if the initial data belongs to the Sobolev space H s (R), with s > 3/2. Namely, we have the following result: Theorem 1.1. Given u 0 ∈ H s (R), s > 3/2, then there exist a maximal time T = T (u 0 ) > 0 and a unique solution u to (1.1) satisfying the initial condition u(0, . Moreover, the solution depends continuously on the initial data, in the sense that the map u 0 → u(·, u 0 ) : ) is continuous and T does not depend on s.
Theorem 1.1 is proved using Kato's approach [27] and its demonstration is presented in Section 2.
In order to enlighten further qualitative analysis of the solutions of (1.1), the next proposition gives us some differential identities.
, its second and third order derivatives exist, v tx = v xt , and where α, β, γ and Γ are constants. Then the following formal identities holds: (1.4) In the very particular case β = γ = 0 and Γ = −α, we have a third identity: The proof of Proposition 1.1 is straightforward and, for this reason, is omitted. However, these identities, when considered on the solutions of the equation (1.2), give us very useful invariants.
is a solution of (1.1) such that u, u x → 0 as x → ±∞, u tx = u xt and its second order derivatives are bounded. Let Then, for any t, we have H 0 (t) = e −λt H 0 (0) and H 1 (t) = e −2λt H 1 (0). In particular, We shall refer to (1.5) generically as energy (functional) and it is nothing but a time dependent H 1 (R)−norm of the solution u (which we shall simply refer as Sobolev norm for convenience). If λ < 0 we easily see that its norm increases as the increasing of t, leading to unbounded solutions on [0, ∞) × R, whereas the case λ > 0 is more interesting, since in this situation the Sobolev norm for any t is bounded from above by the Sobolev norm of the initial data.
Theorem 1.4 can only foresee the emergence of wave breaking of the solutions of (1.2) for limited, but positive, values of λ. We observe that if λ in (1.2) is small, then we can interpret the term λ m as a perturbation of the equation which was studied in [14,15], and also in [5,22,23,36] for specific choices of the parameters α, β, γ and Γ, as already mentioned.
Although we may unriddle the presence of the term λm in (1.1) as a perturbation of (1.7), we cannot underrate it, since its presence brings structural and substantial changes in the behaviour of the solutions of equation (1.1). As we shall show in section Section 3, it implies that if global solutions of (1.2) exist, as well as their corresponding energy functional, then they vanish as t → ∞. Moreover, for λ sufficiently small, the conditions for wave breaking of the solutions (1.2) are unaltered when compared to (1.7), as one can observe comparing our results with those proved in [15] regarding this matter. We discuss our results in Section 6, while in Section 7 we present our conclusions.

Novelty and challenges of this paper
The presence of the cubic and quartic nonlinearities in (1.2) brings some difficulties in the qualitative analysis of the solutions of (1.2) when compared with similar works dealing with (1.7). This is somewhat expected, since the presence of these higher order nonlinearities introduces substantial modifications on the behaviour of the solutions of (1.2) in comparison with (1.7), as one can infer by comparing 1 the results in [6][7][8][9]13,19,28,30,37,39,40] with those in [5,14,15,22,23,36]. In particular, the equation (1.7) does not seem to be integrable unless β = γ = 0, see [14,15]. Turning back to our case, if λβ = 0 or λγ = 0 we have a dramatically different situation to be considered regarding both global existence and wave breaking phenomena. Besides, if one of these conditions is satisfied, then we cannot reduce the analysis of equation (1.2), with λ = 0, to equation (1.7) as it is possible for some dissipative CH type equations, as pointed out in [28].
It was pointed out by Lenells and Wunsch [28] that the analysis of the latter equation can be reduced to the CH equation through the change of variables In our case, if α = −Γ or (β, γ) = (0, 0), then we cannot use the results in [28] to reduce the analysis of equation (1.1) to other previously known.
The clever observation made by Lenells and Wunsch [28] can be applied to a large class of relevant equations in the field of shallow water models and integrable systems, in particular, to some of the references cited above (and perhaps in others in the list) regarding (weakly) dissipative equations. As one can infer from the results in [28], the transformation connecting equations are of the type where p is a certain parameter. The reason for this transformation works is the following (we pay attention only to the scalar cases, but the explanation for systems follows the same argument): The equations considered in [28] are of the form  [28] to their counterparts with λ = 0.
In our case, if α = −Γ we cannot eliminate the term αu x + Γu xxx . On the other hand, if (β, γ) = (0, 0) then we have the presence of cubic and quartic terms. In any situation, we lose homogeneity and, therefore, the elegant transformation introduced in [28] is no longer admissible. Moreover, these higher order terms not only prevent us to invoke the results aforementioned, but also bring difficulties to completely describe conditions for the global existence of solutions for the Cauchy problem (1.2).
Another peculiarity of the present paper is the fact that we construct some invariants for the solutions of the equation with enough decaying at infinity through the establishment of differential identities, as shown in Proposition 1.1. These invariants will provide us estimates on the solutions of the equation that will help us to describe the wave breaking scenario.

Local well-posedness
Here we establish the existence and uniqueness of solutions of the Cauchy problem (1.2). More precisely, we respond questions 1, 2 and 3 in the Introduction of the paper. These results are established at the local level, meaning that we guarantee the existence of T > 0 such that the solution exists on [0, T ) × R.

Overview of functional analysis
We present a short overview of Sobolev spaces, embeddings and mappings between Sobolev spaces. For further readings about these subjects, the reader is referred to [2,21,25,29,38].
If φ is a tempered distribution, its Fourier transform F (φ) and its corresponding inverse are, respectively, given bŷ Given s ∈ R, the Sobolev space of order s is given by H s (R) = Λ −s (L 2 (R)), where L 2 (R)(= H 0 (R)) denotes the usual Hilbert space of the squared integrable functions and Λ s u : The operator Λ s is a unitary isomorphism between H t (R) and H t−s (R), for any t ∈ R. Of particular importance is the case s = 2, in which the operator can be identified with the differential operator Λ 2 := 1 − ∂ 2 x (also known as Helmholtz operator) and its inverse is given by where * denotes the convolution and g(y) = e −|y| /2 is the Green function of the equation We note that H s (R), for each s ∈ R, is a Hilbert space when endowed with the inner product We also note that if u ∈ H 1 (R), then its norm is given by u 2 Some useful results to our purposes are: Proof. See [29] or [38], pages 51 and exercise 6 on page 320, respectively.
As a consequence of the Sobolev Embedding Theorem, if s > 1/2, then u ∈ L ∞ (R). Moreover, if u ∈ H s (R), with s > 1/2 + k, for a certain non-negative number k, then u ∈ C k 0 . In any case, We recall that if E and F are Banach spaces, a mapping f : Proof. See Proposition on page 1065 of [10].
Note that for s > 1/2 and in view of the Sobolev Embedding Theorem, We conclude this subsection with a very useful differential inequality. Proof. See [25], page 56.

Proof of Theorem 1.1
Our main ingredient to prove the local-well posedness of equation (1.1) with initial data u(0, x) = u 0 (x) is the following result, due to Tosio Kato [27].

Lemma 2.5. (Kato's theorem) Let A(u) be a linear operator and consider the problem
(2.1) Assume that the following conditions are satisfied: C1 Let X and Y be reflexive Banach spaces, such that Y ⊆ X and the inclusion Y ֒→ X is continuous and dense. In addition, there exists an isomorphism S : Y → X such that u Y = Su X ; C2 There exist a ball W of radius R such that 0 ∈ W ⊆ Y and a family of operators (A(u)) u∈W ⊆ L(X) such that −A(u) generates a C 0 semi-group in X with e −sA(u) L(X) ≤ e βs , for any u ∈ W , s ≥ 0, for a certain real number β. C3 Let S be the isomorphism in Condition C1. Then B(u) := [S, A(u)]S −1 ∈ L(X). Moreover, there exists constants c 1 and C5 The function f : X → X satisfy the following conditions: The constants mentioned in the conditions in the lemma above depend on the radius R of W , see [19,27,31,37].
where h is given by (2.3).
Let us assume that u and h(u) belongs to H s (R), for a certain s. Since H s (R) is continuously and densely embedded into H s−1 (R), then the right hand side of (2.2) can be viewed as a mapping F : where h(u) is given by (2.3). Moreover, if we define A(u) := (u + Γ)∂ x , we have a linear operator that maps a function v ∈ H s (R) into A(u)v = (u + Γ)v x . If we take s > 3/2, u and v in H s (R), then v x ∈ H s−1 (R) and the algebra property holds, implying that A(u)v ∈ H s−1 (R). Moreover, if s > 1/2, then the function h satisfies the conditions in Lemma 2.3.
We note that (2.4) is an equation in the variables (t, x), but for each fixed x, it is an equation of the type (2.1). We are ready to prove Theorem 1.1.
Proof of Theorem 1.1. It is enough to verify that A(u) := (u + Γ)∂ x and F (u) given by (2.5) satisfy the conditions in Kato's theorem. From the results proved in [14,30] we see that A(u) satisfies all required conditions.
This means that F satisfies the last condition in Kato's theorem if and only if f does, which follows from the results proved in [14,30,31], see also [37].

Time dependent conserved quantities
In this section we shall find some quantities that are conserved on the solutions of the equation in (1.2) for a suitable choice of the initial data. More specifically, we shall show that the quantities e λt R u(t, x)dx and e 2λt R u(t, x) 2 + u x (t, x) 2 dx are constants. This is an immediate consequence of Theorem 1.2 that will play vital role in the investigation of the existence of global solutions and blow-up phenomena.
The relations above show that the result of the integrals has exponential decaying and, in particular, it makes the Sobolev norm · H 1 (R) of the solutions of (1.1) go to 0 if λ > 0 and t → ∞, while for λ < 0 it assures in a very simple, but elegant, way the presence of unbounded solutions.  We observe that in the case λ ≥ 0, then H i (t) ≤ H i (0), i = 1, 2. Therefore, if λ > 0 and u 0 ∈ H 1 (R), equation (1.5) implies that the Sobolev norm of the corresponding solution of (1.1) satisfying u(0, x) = u 0 (x) is time-dependent and decreases along time if u 0 ≡ 0. Then, by the Sobolev Embedding Theorem, u 2 L ∞ (R) ≤ 2H 1 (t) ≤ 2H 1 (0) = u 0 2 H 1 (R) < ∞. Assume that u 0 ≡ 0 and λ < 0. Theorem 1.2 implies that if T = ∞, then |H 1 (t)| → ∞ as t → ∞, meaning that the solution of (1.2) is unbounded.
We summarise the comments above in the next result.
Corollary 3.2. Assume that the initial data u 0 of the problem (1.2) belongs to H s (R), s > 3/2. If u 0 = 0, then the solution is globally defined and u(t, x) ≡ 0, for any (t, x) ∈ [0, ∞) × R and any value of λ. If u 0 ≡ 0 and λ < 0, then the solution is finite on each [0, T ], for any T < ∞, but is not bounded on [0, ∞), while if λ > 0, then the solution u is bounded from above by u 0 H 1 (R) , for any (t, x) ∈ [0, T ) × R.
From Corollary 3.2 we infer the presence of a necessary ingredient to the rising of wave breaking: if λ > 0, then the solution u is spatially bounded for t < T . In Section 4 we shall retake this fact to find conditions to figure out wave breaking of the solutions of the problem (1.2).
Proof of Theorem 1.3. Consider the identity (1.3) with v = u. We can rewrite it as Integrating the expression above over R, we have Solving the ODE above and taking into account that at t = 0 we have m(0, x) = m 0 (x) and obtain the first equality in (1.6).

The last equality is obtained in a similar way by integrating directly (1.3) with v replaced by u.
It remains to be proved the middle equality, but it is a direct consequence of the fact m = u − u xx and u x → 0 as |x| → ∞.

Blow-up scenario
Here we investigate the conditions for the occurrence of wave breaking of the solutions of (1.2). Our main influence here is the text by Escher [19], the works by Constantin and Escher [6][7][8][9], and Mustafa [31]. However, in view of the presence of the function h(u) and the term λu in (2.4), their ideas are not directly applicable to our problem. In fact, we can deal with the term λu following similar procedures presented in [41,42]. The main issue in our case is the term h(u) in (2.4), which we need to control in order to put our problem in a suitable place to be tractable.

Preliminary results
Here we determine when the solution of the equation possesses finite H 3 (R)−norm. We begin with the following result: Theorem 4.1. Let u be a solution of (1.1) with initial data u(0, x) = u 0 (x) and m 0 = u 0 (x) − u ′′ 0 (x). Assume that m 0 ∈ H 1 (R) and there exists a positive constant k such that u x > −k. Then there exist a function σ ∈ C 1 (R) such that u H 3 (R) ≤ σ(t) m 0 H 1 (R). In particular, u does not blow up in finite time.
Proof. We begin with recalling that Λ −2 is a isometric isomorphism between H s and H s+2 . Moreover, since m = u − u xx , then u = Λ −2 m. Our strategy in the present demonstration is to prove that m H 1 (R) ≤ σ(t) m 0 H 1 (R) , for a certain σ ∈ C 1 (R). The result is then obtained from the Let us find the parcels of the right hand side of (4.1). From (1.7) and (2.3) we have Therefore, where we used the relations m, u Deriving (4.2) with respect to x and substituting the result into m x , m tx L 2 (R) yield (4.4) and after some manipulation, we have (4.5) Since 2ab ≤ a 2 + b 2 , for any real numbers a and b, we have We still have the inequality Claim. There exists a constant c 1 depending only on We note that if our claim is true, then h(u) for some positive constant c. In addition, we have (4.7) Substitution of (4.6) and (4.7) into (4.5) reads Using the Gronwall inequality, we conclude that m 2 H 1 (R) ≤ e (5k+c+2λ)t m 0 2 H 1 (R) =: σ(t) 2 m 0 2 H 1 (R) , which is sufficient to have the result proved. To conclude the proof, we must now prove the claim. To do it, note that From the last equality we easily conclude that |h(u)| ≤ |u| sup{|h ′ (r)|, |r| ≤ u L ∞ (R) }. Moreover, we also know that u L ∞ (R) ≤ u H 1 (R) (see, for instance, [31], page 1396). This, combined with the fact that u H 1 (R) ≤ u 0 H 1 (R) implies that |h(u)| ≤ |u| sup{|h ′ (r)|, |r| ≤ u 0 H 1 (R) }. Let c 1 := sup{|h ′ (r)|, |r| ≤ u 0 H 1 (R) }. Then |h(u)| ≤ c 1 |u|, which yields the desired result.
Observe that H 3 (R) ⊆ H s (R), for any s ≤ 3. Moreover, this embedding is dense and continuous. This proves the following Corollary 4.1. Let u 0 ∈ H s , s ≥ 3/2, and u be the corresponding solution to (1.1) with initial data u(0, x) = u 0 (x). Assume that u x > −k, for some positive constant k. Then u H s (R) ≤ σ(t) u 0 H s (R) , for a certain positive function σ ∈ C 1 (R).

Wave breaking criteria
We recall that wave breaking occurs when Corollary 3.1 implies that if λ > 0, then u is bounded, which is a necessary condition for the appearance of wave breaking. On the other hand, Theorem 4.1 assures that if u is a solution of (1.2) and u x is bounded from below, then we do not have wave breaking. The consequence of these facts is: If we want to investigate wave breaking, then we must look for solutions having u x with no lower bound. We observe that if u x (t, x) < 0, then we should replace sup and ∞ by inf and −∞, respectively, in the conditions above whenever we replace |u x (t, x)| by u x (t, x). To understand what we are bound to do, let y(t) := inf{u x (t, x), x ∈ R}, which we simply write as y(t) := inf and the function y is almost everywhere differentiable (a.e) in (0, T ), with y ′ (t) = v tx (t, ξ(t)) a.e. on (0, T ).
Then h(u) H 1 (R) ≤ κU 0 , for a certain positive constant κ depending on α, β, γ and Γ. Moreover, From the comments above, Proposition 5.1 and equation (4.10) we have the following inequality Suppose that u 0 ∈ H 3 (R) be an initial data of (1.2) such that for some constant θ > 0 and some point x 0 ∈ R. Since y(t) = inf Suppose that we might be able to find ǫ such that If we could choose θ ≤ 2/(1 + 12κ), then the inequality (4.13) would read We are bound to find conditions for the occurrence of wave breaking. We only need two very technical observations that will make easier the proof of Theorem 1.4. satisfy the condition (4.14) for some θ > 0, let us define (4.16) where ǫ 0 is given by (4.16). If λ ∈ (0, λ 0 ), then Under the conditions of the theorem, we note that y(t) = inf x u x (t, x) satisfies the inequality (4.15).

Comments on the limitations to ensure global existence of solutions
In this section we show the limitations to completely describe the scenario for global existence. We begin with a technical result proving the existence of a diffeomorphism.
) be a solution of (1.1). Then the problem has a unique solution q(t, x) and q x (t, x) > 0, for any (t, x) ∈ [0, T ) × R. Moreover, q(t, ·) is an increasing diffeomorphism of the line.
Proof. We prove that the system has a unique solution q(t, x) and that q x (t, x) > 0, for any (t, x) ∈ [0, T ) × R. The proof that this function is a diffeomorphism is the same as for the CH equation and it can be found in [9], Theorem 3.1.
Since u ∈ C 1 ([0, T )H 2 (R)) and H 2 (R) ⊂ C 1 (R), then u ∈ C 1 ([0, T ) × R, R), which means that problem (5.1) has a unique solution. If we differentiate (5.1) with respect to x, and noticing that ∂ x u(t, q) = u x q x , we have another IVP, given by The solution of (5.2) is which completes the proof.
We note that Theorem 4.1 describes a sufficiency condition for the global existence of solutions, but it does not give us any information about whether this condition is satisfied.
Proof. Since u = p * m, where * denotes the convolution and p = e −|x| /2, we have Differentiating this representation of u with respect to x gives x e −ξ m(t, ξ)dξ.
Since sgn m = sgn m 0 , then m(t, x) ≤ 0 if x ≤ q(t, x 0 ) and m(t, x) ≥ 0 if x ≥ q(t, x 0 ), where q is the function in Proposition 5.1. Therefore, x e −ξ m(t, ξ)dξ As a consequence, if x ≥ q(t, x 0 ), then u x (t, x) ≥ −u(t, x). On the other hand, a similar calculation reads These two facts are enough to assure that u We Proof. Differentiating m(t, q(t, x))q 2 x (t, x) with respect to t and using (4.2), we conclude that d dt which is a linear ODE to mq 2 x . Integrating (5.4) and taking (5.1) and (5.2) into account we conclude that its solution is (5.3). A possible direction to have a complete description of the existence of global solutions is the following: Let h(u) be the function given by (2.3). From (4.5) we have m(t, q(t, x)) q 2 x (t, x) = e −λt m 0 (x) + t 0 e λ s q 2 x (s, x)∂ x h(u(s, q))ds .
In line with the remarks above, we have the following corollary. Proof. It follows from (5.5) that sgn (m) = sgn (m 0 ) and the conclusion is a consequence of theorems 4.1 and 5.1.

Discussion
Very recently [5,22,23,36], an equation of the type (1.1) was deduced as a model for shallow water waves with Coriolis effect. The mentioned equation has its coefficients depending on physical parameters related to the rotation of the Earth. On the other hand, in [14] we considered (1.1) with λ = 0 and we investigated it from a complementary point of view (taking the results in [5,22,23,36] into account). This equation revealed to be mathematically very rich, as one can see by the multitude of travelling waves [14,23] it possesses, local well-posedness [5,14] and wave breaking [15] The main difference of our results and those established in [14,15] is the presence of the term λm in (1.1) or λu in (2.4). From a different perspective, we can also argue that the main difference between the problem we treated here with those treated in [32,34,35,[41][42][43] is the presence of the cubic and quartic nonlinearities in (1.1).
In some parts of the text we mentioned dissipation. Let us explain the term and why this is our case. Let u 0 be an initial data of (1.2) such that 0 = u 0 ∈ H s (R), with s > 3/2, and λ > 0. From Theorem 1.2 we have the energy H 1 (t) = e −2λt H 1 (0), and H 1 (0) = u 0 2 H 1 (R) /2 > 0. Then we can easily infer dH 1 (t) dt = −2λH 1 (t) = −2λe −2λt H 1 (0) < 0. (6.1) From (6.1) we observe that H 1 (t) decreases with time, which means that the energy is not conserved along time, or better, we have loss, or dissipation, of energy. Moreover, we observe that the energy of the solution is a monotonic decreasing function of t.
The presence of the function h(u) in (2.4) or, more precisely, the cubic and quartic nonlinearities in (1.1), brings some complexity to the problem when compared with similar results of CH and DGH equations. For example, the condition we found for the existence of wave breaking are affected by the values of the parameters, as one can see by the range of θ in the condition (4.14) given by Theorem 1.4. Moreover, the value of ǫ 0 also depends on powers of the norm u 0 H 1 (R) , see (4.16), as well as the upper bound to λ, as shown in (4.17). If u 0 H 1 (R) ≤ 1 then (4.16) and (4.17) reduce to ǫ 0 = θ 2 u ′ 0 (x 0 ) 2 − u 0 H 1 (R) θ 2 u ′ 0 (x 0 ) 2 and λ 0 = − y(0) 4 3. We also obtain sufficient condition for the appearance of wave breaking of the solutions of (1.2), see Theorem 1.4.