Blow-up to a shallow water wave model including the Degasperis-Procesi equation

1.   Preliminary

This work is to probe blow-up feature of the equation

where constants $k\in\mathbb{R}, \alpha\in\mathbb{R}$ and $m > 0$. Equation (1.1) depicts the motion of shallow water waves (see Constantin and Lannes ^[2]). Actually, the shallow water wave model deduced in ^[2] includes Eq (1.1).

Setting $k = 0, \; \alpha = 0$ and $m = 4$, Eq (1.1) is turned into the famous Degasperis-Procesi (DP) model ^[6]

The global weak solutions and assumptions to cause the wave breaking of solution for Eq (1.2) are explored in Escher et al. ^[7]. For certain partial differential equations, their solution remains bounded, but its derivative about space variable tends to infinite at the blow-up time. This phenomena is called wave breaking of solutions (see Constantin ^[2,3]). The dressing method is employed in Constantin and Ivanov ^[4] to investigate dynamical features of the DP model (1.2). The global strong solutions and wave breaking phenomena for the DP model are explored with certain functional spaces in ^[14,24]. The large time asymptotic features of the periodic entropy (discontinuous) solutions for DP equation is considered in ^[5]. Lundmark and Szmigielski ^[15] study the multi-peakon solutions of the Eq (1.2) (also see Matsono ^[12,19]). Lenells ^[18] finds out many traveling wave solutions of the DP model. Periodic and solitary wave solutions to the DP model are classified in Vakhnenko and Parkes ^[22]. Two conservation laws to Eq (1.2) are utilized to investigate the stability of peakons in Lin and Liu ^[16]. Lai and Wu ^[17] study $L^1$ local stability for a shallow water wave equation including DP equation endowed with certain conditions. The infinite propagation speed of DP model is discussed in Henry ^[10]. Akinyemi et al. ^[1] apply an efficient numerical simulation method to study the coupled nonlinear Schr$\ddot{o}$dinger-Korteweg-de Vries and Maccari systems. Utilizing the properties of fractional operators and the numerical computational techniques, Veeresha et al. ^[23] investigate the shallow water forced Korteweg-De Vries model associated with critical flow over a hole (see ^[13]). For nonlinear models relating to Eq (1.2), we refer the reader to ^{[8,9,11,20,21]} and the references therein.

For Eq (1.1) endowed with $v(0, x) = v_{0}(x)\in H^s(\mathbb{R}), s > \frac{3}{2}$, we derive that

which leads to

The objective of this work is to study the shallow water wave Eq (1.1), which generalizes the famous Degasperis-Processi model. We find that the wave breaking of the solutions for Eq (1.1) behaves the same structure as that of the DP model (see ^[8,20]). The novelty in our work is that we derive $L^2(\mathbb{R})$ conservation law (1.3), which takes a key role to derive several bounded estimates of solutions for Eq (1.1). For any constant $\alpha$, we find arguments to support the bounded property of $\parallel v(t, \cdot)\parallel_{L^\infty(\mathbb{R})}$. For blow-up time $T$, we deduce that when the solution of Eq (1.1) blows up, namely, $\lim\limits_{t\rightarrow T}\parallel v(t, \cdot)\parallel_{H^2} = \infty$ if and only if the space derivative of the solution tends to minus infinite. Here we state that the main technique used in this work is the classical energy estimate methods.

In Section 2, we give the local well-posedness of solution for Eq (1.1) and derive the conservation law (1.3). Several Lemmas about the bound property of the solutions are established. Section 3 provides conditions imposing on the initial value to discuss the wave breaking for Eq (1.1). Conclusions are summarized in Section 4.

2.   Basic $L^\infty$ estimate and several lemmas

We write the Cauchy problem for Eq (1.1) in the form

where $k, \alpha$ are constants, constant $m > 0$ and $v_{0}(x)\in H^s(\mathbb{R})$ with $s > \frac{3}{2}$.

Utilizing $\Lambda^{-2} = (1-\frac{\partial^2}{\partial x^2})^{-1}$ and $\Lambda^{-2}h(x) = \frac{1}{2}\int_Re^{-|x-z|}h(z)dz$ for every $h\in L^p(\mathbb{R}) (1\leq p\leq\infty)$, we acquire

In order to discuss the blow-up of solution, we introduce the local existence result for problem (2.1).

Lemma 2.1. (^[2]) Assume $s > \frac{3}{2}$ and $v_{0}(x)\in H^s(\mathbb{R})$. Then, problem (2.1) exists a unique solution $v$ satisfying

where $T = T(v_{0})$ stands for the maximal existence time of $v(t, x)$.

Lemma 2.2. Assume $m > 0$, $y = v-\frac{\partial^2v}{\partial x^2}$ and $W = (m-\frac{\partial^2}{\partial x^2})^{-1}v$. Let $s > \frac{3}{2}$, $v_{0}\in H^s(\mathbb{R})$. If $v$ satisfies problem (2.1), then,

Moreover,

which is equivalent to

Proof. We have $v = mW-W_{xx}$ and $W_{xx} = mW-v$. Using (1.1) and integration by parts yields

which together with the Parserval identity, we obtain (2.3). Using (2.3) derives (2.4).

The conservation law in Lemma 2.2 takes a key role to establish the bounds of solutions for problem (2.1).

Lemma 2.3. Provided that $v_{0}(x)\in H^s(\mathbb{R})$ $(s > \frac{3}{2})$, then,

Proof. Setting $\Lambda^{-2}v = V$, we have

which together with integration by parts, yields

and

Using Eqs (2.5)–(2.7) ends the proof.

Utilizing Lemma 2.2, we can establish the estimates about the operator $\Lambda^{-2}$.

Lemma 2.4. Provided that $v(t, x)$ satisfies (2.2) and $v_{0}(x)\in H^s(\mathbb{R})$ $(s > \frac{3}{2})$, then,

where $c_m = \max\Big(\sqrt{m}, \sqrt{\frac{1}{m}} \Big)$.

Proof. Utilizing $\int_{\mathbb{R}}e^{-|x-y|}dx = 2$, we obtain

The Tonelli theorem and Lemma 2.2 ensure that

Applying

we acquire

and

Using Eqs (2.8)–(2.10) arises

Applying (2.8)–(2.11) ends the proof.

Lemma 2.5. Assume $v\in H^s(\mathbb{R})$ with $s\geq3$. Then,

Proof. Utilizing integration by parts arises

which leads to the first identity in (2.12). Since

which deduces that the second identity in (2.12) holds.

For $t\in[0, T)$, we consider the problem

Lemma 2.6. Assume that $T$ and $v_{0}\in H^s(\mathbb{R}), s\geq 3$ are defined as in Lemma 2.1. Then, problem (2.13) exists a unique solution $q$ such that $q\in C^1([0, T)\times \mathbb{R}, \mathbb{R})$ and $q_x(t, x) > 0$ for $(t, x)\in [0, T)\times\mathbb{R}$.

Proof. If $(t, x)\in [0, T)\times \mathbb{R}$, using the conclusion in Lemma 2.1, we have $v_x\in C^2(\mathbb{R})$ and $v_t\in C^1[0, T)$. Thus, we know that $v(t, x)$ and $v_x(t, x)$ are Lipschitz continuous with respect to $x$ and $t$. Using the well-posedness of ordinary differential equation, we obtain that (2.13) exists a unique $q\in C^1([0, T)\times \mathbb{R}, \mathbb{R})$.

From (2.13), we have

which results in

For every $T' < T$, we obtain $\sup\limits_{(t, x)\in [0, T')\times\mathbb{R}}|v_x(t, x)| < \infty$. Thus, there must exist a constant $C_0 > 0$ to ensure $q_x(t, x)\geq e^{-C_0t} > 0$.

The isomorphic property about $q(t, x)$ is very important to prove the following Lemma.

Lemma 2.7. Provided that $t\in [0, T)$, $v_{0}\in H^s(\mathbb{R})$, $s > \frac{3}{2}$, then,

in which $c_m = \max\Big(\sqrt{m}, \sqrt{\frac{1}{m}} \Big)$.

Proof. Setting $\eta(x) = \frac{1}{2}e^{-\mid x\mid}$, we obtain $\Lambda^{-2}h = \eta\star h$ with $h\in\; L^p(\mathbb{R})\; (1\leq p\leq \infty)$. The density arguments in ^[14] arrow us to only consider Lemma 2.7 for $s = 3$. For $v_{0}\in H^3(\mathbb{R})$, Lemma 2.1 ensures $v\in C([0, T), H^3(\mathbb{R}))\cap C^1([0, T), H^2(\mathbb{R}))$. Making use of (2.2) yields

Using the H$\ddot{o}$rlder inequality yields

We have

and

Applying (2.14) and (2.17) yields

which together with (2.15) and (2.16), we get

From (2.18), we obtain

Thus,

which together with Lemma 2.6 ends the proof.

3.   Blow-up scenario

Provided that the maximal time of existence $T > 0$ for problem (2.2) is finite and $v_{0}(x)\in H^3(\mathbb{R})$, Lemma 2.1 guarantees existence $v(t, x)\in C([0, T);\; H^3(\mathbb{R}))\cap C^1([0, T);\; H^{2}(\mathbb{R}))$. When $\alpha = 0, m = 4$, it is derived in ^[2,14] that $\parallel v\parallel_{L^\infty(\mathbb{R})}$ is bounded as $t$ tends to $T$. For an arbitrary $\alpha$ in Eq (1.1), Lemma 2.7 ensures the bounded feature of $\parallel v\parallel_{L^\infty(\mathbb{R})}$. We shall verify that the blow-up occurrence of Eq (1.1) is analogous to the wave breaking phenomena of the DP model.

Theorem 3.1. Let $v_{0}(x)\in H^3(\mathbb{R}), \; m > 0$ and $T > 0$ be defined as in Lemma 2.1. Then, $\lim\limits_{t\rightarrow T}\parallel v(t, \cdot)\parallel_{H^2} = \infty$ is equivalent to

Proof. Lemma 2.1 ensures $v(t, x)\in C([0, T), \; H^3(\mathbb{R}))\cap C^1([0, T), \; H^2(\mathbb{R}))$.

From system (2.2), we have

in which Lemma 2.3 is used.

From (2.2), we obtain

Multiplying Eq (3.3) by $v_x$ and using Lemmas 2.3 and 2.5 yield

Differentiating (3.3) about $x$ gives rise to

Multiplying (3.5) by $v_{xx}$, using integration by parts and Lemmas 2.3 and 2.5, we obtain

Applying (3.2), (3.4) and (3.6) yields

Provided that for any $(t, x)\in[0, T)\times\mathbb{R}$ and $\lim\limits_{t\rightarrow T}\parallel v(t, \cdot)\parallel_{H^2(\mathbb{R})} = \infty$, we assume that there is a constant $C > 0$ satisfying

Employing (3.7)–(3.8) and Lemma 2.4 gives rise to

Letting

and using (3.9), we obtain

Utilizing the Gronwall inequality yields

which leads to $v(t, x)\in H^2(\mathbb{R})$, meaning that (3.8) is wrong. On the other hand, if (3.1) holds, using the inequality $\parallel v_x\parallel_{L^\infty}\leq \parallel v(t, \cdot) \parallel_{H^2(\mathbb{R})}$, we obtain $\lim\limits_{t\rightarrow T}\parallel v(t, \cdot) \parallel_{H^2(\mathbb{R})} = \infty$.

Theorem 3.2. Let $s\geq 3$, $v_x(0, 0) < 0$, $\alpha = k$ and $m\geq 1$. Provided that $v_{0}(x)$ is an odd function and $v_{0}(x)\in H^s(\mathbb{R})$, then, solution $v(t, x)$ of system (2.2) blows up at time $T$ and $T$ is bounded above $-\frac{1}{v_x(0, 0)}$.

Proof. Employing Lemma 2.1 ensures the existence $v\in C([0, T);\; H^3(\mathbb{R}))\cap C^1([0, T);\; H^{2}(\mathbb{R}))$.

The symmetry $(v, x)\rightarrow (-v, -x)$ holds for system (2.2) if $v_{0}(x)$ is an odd function. Using system (2.2) and the assumption in Theorem 3.2 yields

Using (3.3) gives rise to

Using $\Lambda^{-2}v^2\geq 0$, $m\geq 1$, setting $Y(t) = v_x(t, 0)$, from (3.10)–(3.11), we deduce that

which yields

Thus,

The proof is finished.

4.   Conclusions

For shallow water wave model (1.1) with an arbitrary constant $\alpha$ and Degasperis-Procesi equation (1.2), if the initial value $v_{0}\in H^3(\mathbb{R})$, both of them possess the wave breaking feature. Namely, their solutions remain bounded and their slopes become infinite when their solutions blow up at finite time $T$. It is concluded that the shallow water wave Eq (1.1) and DP model behave the same blow-up structure in certain sense. The further question is to find other simple conditions imposing on the initial data to ensure that the wave breaking happens for Eq (1.1). Using the numerical simulation methods to discover the dynamical characteristics of the solutions for certain inhomogeneous boundary conditions for Eq (1.1) also needs to be investigated.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgements

This work is supported by the Younth Scientific Research Project of Yili Normal University (Grand No. 2023YSQN002) and the ‘14th Five-Years’ Key Discipline of Xinjiang Uygur Autonomous Region.

Conflict of interest

The authors declare no conflict of interest.