Abstract
In this paper, we investigate the Cauchy problem for the N – abc family of Camassa-Holm type equation with both dissipation and dispersion. Furthermore, we establish the blow-up result of the positive solutions in finite time under certain conditions on the initial datum. This result complements the early one in the literature, such as [E. Novruzov, Blow-up phenomena for the weakly dissipative Dullin-Gottwald-Holm equation, J. Math. Phys. 54 (2013), no. 9, 092703, DOI 10.1063/1.4820786] and [Z.Y. Zhang, J.H. Huang, and M.B. Sun, Blow-up phenomena for the weakly dissipative Dullin-Gottwald-Holm equation revisited, J. Math. Phys. 56 (2015), no. 9, 092701, DOI 10.1063/1.4930198].
1 Introduction
Differential equations and dynamical modeling have attracted some attention from many researchers as a result of their potential applications in fields of biology [3, 4, 5, 6], physics [7, 8, 9], engineering [10, 11], information technology and so forth [12, 13, 14]. Since the seminal work by Camassa and Holm [15], Camassa-Holm type equations have been intensively investigated. In this paper, we consider the Cauchy problem for the N – abc family of Camassa-Holm type equation with both dissipation and dispersion
where N ∈ Z+, N ≥ 2, k, λ ≥ 0, and k, λ are dissipation and dispersion coefficients respectively. a, b, c are positive constants and a = b + c.
When c = 1, a = b + 1, k = λ = 0, the first equation of (1.1) becomes
Eq. (1.2) was first investigated by Himonas and Holliman [16] and they proved the local well-posedness and the nonuniform dependence of its Cauchy problem in Sobolev space Hs with s >
It is important to note that (1.1) is an evolution equation with (N + 1)-order nonlinearities and includes three famous integrable dispersive equations: the Camassa-Holm (CH) equation, the Degasperis-Procesi (DP) equation and the Novikov equation (NE).
As c = 1, N = 1, b = 2, a = 3, k = λ = 0, (1.1) becomes the well-known CH equation. The local well-posedness of Cauchy problem of the CH equation has extensively been investigated in [20]. It was shown that there exist global strong solutions to the CH equation [20] and finite time blow-up strong solutions to the CH equation [20, 21]. The existence and uniqueness of global weak solutions to the CH equation were studied in [22].
As c = 1, N = 1, b = 3, a = 4, k = λ = 0, (1.1) reads the DP equation in [23]. It is another integrable peakon model with quadratic nonlinearity, but with 3 × 3 Lax pairs [24]. The local well-posedness, global existence and blow-up phenomena of the DP equation was studied in [25, 26, 27, 28].
As c = 1, N = 2, b = 3, a = 4, k = λ = 0, (1.1) reads the NE in [29], which is also integrable peakon model with 3 × 3 Lax pairs and the peakon solution u(x, t) =
It is well known that it is difficulty to avoid energy dissipation in a real world. Thus it is reasonable to study the model with energy dissipation in propagation of nonlinear waves, see [35, 36, 37, 38]. Recently, Wu and Yin [39] investigated the blow-up, blow-up rate and decay of solutions to the weakly dissipative periodic CH equation (i.e (1.1) with N = 1, c = 1, b = 2, a = 3, k = 0). Thereafter, they also studied the blow-up and decay of solutions to weakly dissipative non-periodic CH equation (i.e (1.1) with N = 1, c = 1, b = 3, a = 4, k = 0) [40]. Hu and Yin [41] investigated the blow-up, blow-up rate of solutions to weakly dissipative periodic rod equation. Later on, Hu [42] discussed the global existence and blow-up phenomena for a weakly dissipative periodic two component CH system. Zhou, Mu and Wang [43] considered the weakly dissipative gCH equation (i.e (1.1) with c = 1, a = b + 1, k = 0). Recently, Novruzov [1] studied the Cauchy problem for the weakly dissipative Dullin-Gottwald-Holm (DGH) equation (i.e (1.1) with N = 1, c = 1, b = 2, a = 3) and establish certain conditions on the initial datum to guarantee that the corresponding positive strong solutions blow up in finite time. The same equation for arbitrary solution has been considered in [44]. Authors showed the simple conditions on the initial data that lead to the blow-up of the solutions in finite time or guarantee that the solutions exist globally. Later on, Zhang et al. [2] improved the results of [1]. In [45], Novruzov extended the obtained “blow-up” result to the DGH equation under some conditions on the initial data. This issue is extensively studied, e.g. in [46, 47, 48].
2 Preliminaries
In this section, we recall some useful results in order to achieve our aim.
Let us first present the local well-posedness of Cauchy problem for (1.1). Thus, we can rewrite (1.1) in the equivalent form. Let y = u – uxx. Then (1.1) becomes
Notice that G(x) =
where
The local well-posedness of Cauchy problem for (1.1) with the initial data u0(x) ∈ Hs, s >
Lemma 2.1
Given u0(x) ∈ Hs, s >
is continuous and the maximal time of existence T > 0 can be chosen to be independent of index s.
The following lemma gives necessary and sufficient condition for the blow-up of the solution.
Lemma 2.2
Given u0(x) ∈ Hs, s >
Proof
Indeed, the above result follows by standard manner in [49]. Assume u0 ∈ Hs for some s ∈ N, s ≥ 2. Multiplying both sides of the first equation of (2.1) by 2y = 2u – 2uxx and integrating by parts with respect to x, we get
that is,
which implies the following result:
∙ if ∥u∥L∞ and ∥ux∥L∞ are bounded on [0, T), that is, there exists a positive constant K, such that ∥u∥L∞, ∥ux∥L∞ ≤ K.
Then, based on the above arguments and noticing (2.3), we have
where C is a positive constant. On one hand, we observe that
Using the Gronwalls inequality, from (2.4) and (2.5), we obtain
which implies that the H2-norm of the solution to Eq. (2.1) does not blow up in a finite time. On the other hand, by Sobolev’s imbedding theorem, if
Then the solution will blow up in a finite time. By density argument, we know that Lemma 2.2 holds for all s >
Remark 2.1
When N = 1, 2, we refer to the proof of Theorem 3.1 in [30]. In this sense, when N ≥ 2, this result improves their results.
Remark 2.2
Lemma 2.2 covers Lemma 5.1 in [50].
Consider now the following initial value problem
where u(x, t) is the corresponding strong solution to (1.1).
After simple computations and solving (2.6), we get the following lemma.
Lemma 2.3
Let u0(x) ∈ Hs, s > 3, and let T > 0 be the maximal existence time of the solution u to (1.1). Then, we have
which implies
In particular, if N = 2b, we have
Proof
It follow from (1.1) and (2.1) that
which implies
Thus, setting ξ = q(t, x), we arrive at
Obviously, letting N = 2b leads to e–λt∥y0∥L2 = ∥y∥L2. This completes the proof of Lemma 2.3.
Finally, let us now give the following lemma which will be used in the sequel.
Lemma 2.4
Let u0(x) ∈ Hs, s ≥ 3, c =
Proof
Multiplying both sides of Eq. (2.1) by u, we get
Noticing that
and
we obtain
which implies the desired result in the lemma.
3 Main result
We are now position to state our main result.
Theorem 3.1
Let b = c(N + 1). u0(x) ∈ Hs, s ≥ 3, is such that y0 =
is satisfies. Then the corresponding positive solution u(t, x) to (1.1) blows up in finite time. Here
Proof
We shall give the proof by contradiction. We assume that it is not true and solutions exist globally. That is, there exists constant C such that ux ≥ –C (due to Lemma 2.2).
Observing that u = G * y with G(x) =
So, we have
For t ∈ [0, T), q(t, ⋅) is the increasing diffeomorphism of the line. From Lemma 2.3, we deduce that y(t, x) ≤ 0, x ≥ q(t, x0). Hence, we conclude that –ux(t, x) ≥ u(t, x) ≥ 0, for x ≥ q(t, x0). Due to ux ≥ –C, we get
Differentiating (2.2) with respect to x and noticing that
we have
Multiplying (3.3) by (1 – α)e–x(–ux)α(1 < α < 2) and integrating over (q(t, x0), θ) (θ < ∞), we have
Next, we shall estimate Ij(j = 1, 2, ⋅⋅⋅, 7) in (10). First, integrating by parts, we have
By (3.2) and ∥G∥L2 ≤ ∥∂xG∥L1 = 1, we get
Substituting (3.5)-(3.9) into (3.4) and observing that
we conclude
Let β = k – λ (α – 1). By c =
where
Setting
By (2.6), we have
Note that θ can be taken to be a sufficient large. Due to
and
and by Jensen’s inequality, we arrive at
Thus, we conclude that
Multiplying both sides of (3.11) by
That is,
Integrating with respect to t, we have
Noticing that θ can tend to ∞, we obtain
where L(t) =
From q(τ, x0) = (c
where
On the other hand,
and by condition (3.1). Thus, we get
Hence, from (3.13), we have L → 0 as
That is, there exists a sequence (tn, xn) such that –ux(tn, xn) → ∞ as t → T͠ which contradicts with (3.2). Thus, our main result is completed.
Acknowledgement
This work was supported by Scientific Research Fund of Hunan Provincial Education Department Nos.18A325, 17A087, 17B113, 17C0711, NNSF of China Grant Nos. 11671101, 61863010, 11926205, Natural Science Foundation of Hainan Province No. 119MS036. Also, this work was partially supported by Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering of Changsha University of Science and Technology Grant No. 018MMAEZD191 and NNSF of China Grant Nos. 71471020, 51839002.
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