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The dynamic properties of solutions for a nonlinear shallow water equation


The local well-posedness for the Cauchy problem of a nonlinear shallow water equation is established. The wave-breaking mechanisms, global existence, and infinite propagation speed of solutions to the equation are derived under certain assumptions. In addition, the effects of coefficients λ, β, a, b, and index k in the equation are illustrated.

1 Introduction

We aim to consider the problem

$$\begin{aligned} \textstyle\begin{cases} v_{t}-v_{xxt} +\beta (v_{x}-v_{xxx})+\lambda (v-v_{xx})+(a+b)v^{k}v _{x} \\ \quad = bv^{k-1}v_{x}v_{xx}+av^{k }v_{xxx} , \\ v(0,x)=v_{0}(x). \end{cases}\displaystyle \end{aligned}$$

Here \((t,x)\in \mathbb{R}^{+}\times \mathbb{R}\), \(v(t,x)\) is fluid velocity of water waves, \(\lambda \in \mathbb{R}^{+}\), \(\beta \in \mathbb{R}\), \((a,b)\in \mathbb{R}^{2}\), k is a positive integer, \(\beta (v-v_{xx})\) is the diffusion term, \(\lambda (v-v_{xx})\) is the dissipative term, \(v_{0}\in B_{p,r}^{s}(\mathbb{R})\ (s>\operatorname{max}(1+ \frac{1}{p},\frac{3}{2} ))\).

Recently, the Camassa–Holm (CH) equation

$$ v_{t}-v_{xxt}+\beta v_{x} +3vv_{x}=2v_{x}v_{xx}+vv_{xxx} $$

has attracted much attention. Equation (1.2) admits blow-up phenomena. Replacing v with \(v+\beta \) in Eq. (1.2), we obtain

$$ v_{t}-v_{xxt}+\beta (v_{x}-v_{xxx}) +3vv_{x}=2v_{x}v_{xx}+vv_{xxx}. $$

Taking \(k=1, \lambda =0, a=1, b=2\) in (1.1) gives rise to the Cauchy problem of Eq. (1.3). The solution v to Eq. (1.2) is viewed as a perturbation near β (see [20]). The properties of solutions to the problem with dispersion and dissipative terms are discovered in [15]. Mi et al. [12] investigate the dynamical properties for a generalized CH equation. For a related study of the CH equation and other related partial differential equations, one may refer to references [3, 7, 11, 14, 16].

Taking \(k=1, \lambda =\beta =0, a=1, b=3 \) in (1.1) yields the Degasperis–Procesi equation

$$\begin{aligned} v_{t}-v_{xxt}+4vv_{x} =3v_{x} v_{xx}+vv_{xxx}. \end{aligned}$$

The formation of singularity for solutions to (1.4) is discovered in [17]. Lai and Wu [10] study the local well-posedness for the Cauchy problem of

$$ v_{t}-v_{xxt}+\beta v_{x}+(a+b)v_{x} =bv_{x} v_{xx}+avv_{xxx}, $$

where \(\beta , a, b \in \mathbb{R}\).

Taking \(k=2, \lambda =\beta =0, a=1, b=3 \) in (1.1), we obtain the Novikov equation

$$\begin{aligned} v_{t}-v_{xxt}+4v^{2}v_{x} =3vv_{x} v_{xx}+v^{2} v_{xxx}. \end{aligned}$$

Guo [4] studies the persistence properties of solutions to the CH-type equation. Fu and Qu [2] discover blow-up of solutions to Eq. (1.6) in \(H^{s}(\mathbb{R})\ (s>\frac{5}{2})\). The peakon solutions to the Novikov equation are established in [6].

Himonas and Thompson [8] discover persistence properties for solutions if \(\lambda =\beta =0, a=1\) in (1.1). The behaviors of solutions [5], global existence of solutions for \(a=1\) [9], and infinite propagation speed of solutions [9, 19] to the problems are investigated. We extend parts of results in [9, 10, 13, 18, 19].

Let \(s\in \mathbb{R}, T>0, p\in [1,\infty ]\) and \(r\in [1,\infty ]\). Thus we set

$$\begin{aligned} E_{p,r}^{s}(T)= \textstyle\begin{cases} C([0,T];B_{p,r}^{s}(\mathbb{R}))\cap C^{1}([0,T];B_{p,r}^{s-1}( \mathbb{R})),& 1\leq r< \infty , \\ L^{\infty } ([0,T];B_{p,\infty }^{s}(\mathbb{R}))\cap \operatorname{Lip}([0,T];B _{p,\infty }^{s-1}(\mathbb{R})),& r=\infty. \end{cases}\displaystyle \end{aligned}$$

Letting \(P_{1}(D)=-\partial _{x}(1-\partial _{x }^{2})^{-1}, P_{2}(D)=(1- \partial _{x }^{2})^{-1} \), problem (1.1) is turned into

$$ \textstyle\begin{cases} v_{t}+(av^{k}+\beta )v_{x}=P_{1}(D)[\frac{b}{k+1}v^{k+1}+ \frac{3ak-b}{2}v^{k-1}v_{x}^{2}] \\ \phantom{v_{t}+(av^{k}+\beta )v_{x}=}{}+P_{2}(D)[\frac{(k-1)(ak-b)}{2} v^{k-2}v_{x}^{3}- \lambda v ], \\ v(0,x)=v_{0}(x). \end{cases} $$

Now we summarize the main results in this paper.

Theorem 1.1

Suppose \(1\leq r, p \leq \infty \), \(v_{0} \in B_{p,r}^{s}(\mathbb{R})\ (s>\operatorname{max}(1+\frac{1}{p}, \frac{3}{2} ))\). Then solution \(v \in E_{p,r}^{s}(T) \) to problem (1.1) is locally well-posed for certain \(T>0\).

Theorem 1.2

Suppose \(1\leq r, p\leq \infty \), \(v_{0} \in B_{p,r}^{s}(\mathbb{R})\ (s>\operatorname{max}(1+\frac{1}{p}, \frac{3}{2} ))\), \(t\in [0,T]\). Then a solution v to problem (1.1) blows up in finite time if and only if

$$ \int _{0}^{t} \bigl(1+ \Vert v_{x} \Vert _{L^{\infty }}\bigr)^{k} \,d\tau = \infty. $$

Theorem 1.3

Suppose \(b=a(k+1)\) and \(v_{0} \in H^{s}( \mathbb{R})\ ( s> \frac{3}{2})\), \(t\in [0,T]\). Then a solution v to problem (1.1) blows up in finite time if and only if

$$ \lim_{t\rightarrow T^{-}}\inf_{x\in \mathbb{R}}v_{x}(t,x)=- \infty. $$

Theorem 1.4

Suppose \(b=a(k+1)\) and \(v_{0} \in H ^{s}( \mathbb{R})\ (s\geq 2)\) satisfies \(\Vert v_{0}-v_{0,xx} \Vert _{L^{2}}< \frac{ 4\lambda }{|a|(k+2) \Vert v_{0} \Vert ^{k-1}_{H^{1}}}\). Then there exists a global solution to problem (1.1) in \(H ^{s}(\mathbb{R})\ (s\geq 2)\).

Theorem 1.5

Assume \(v_{0}\in H^{s}(\mathbb{R})\ (s\geq 2)\), \(n_{0}(x)=v_{0}-v_{0,xx}\neq 0\) for all \(x\in \mathbb{R}\), \(\Vert n_{0} \Vert _{L^{2}}< (\frac{2^{k+1}\lambda }{|ak-2b|})^{ \frac{1}{k}}\) and \(b\neq \frac{ak}{2}\). Then a solution v to problem (1.1) is global in \(H ^{s}(\mathbb{R})\ (s\geq 2)\).

Theorem 1.6

Assume \(a>0\) and let \(v_{0}\in H^{s}( \mathbb{R})\ (s>\frac{5}{2})\) be compactly supported in \([a_{0},b_{0}]\), \(t\in [0,T]\). Suppose k is a positive odd number and \(b=ak\), or \(k=1, 0< b<3a\). Then, the solution \(v(t,x) \) to (1.1) satisfies

$$ v(t,x)=\frac{1}{2}L_{+}(t)e^{-x}\quad \textit{for } x\geq p(t,b_{0}),\qquad v(t,x)=\frac{1}{2}L_{-}(t)e^{x} \quad\textit{for } x\leq p(t,a_{0}), $$

where \(L_{+}(t)\) and \(L_{-}(t)\) are continuous non-vanishing functions given in (4.1). What is more, \(L_{+}(t)>0\), \(L_{-}(t)<0\) for \(t\in [0,T]\). In particular, if \(k=1 \), \(b=2a \) or \(b=\frac{a}{2} \), then \(L_{+}(t)\leq C_{3} e^{(\beta -\lambda )t}\) and \(|L_{-}(t)| \leq C_{4} e^{-(\beta +\lambda )t}\).

Remark 1.1

Problem (1.1) is local well-posed in \(B_{p,r} ^{s}(\mathbb{R})\ (s>\operatorname{max}(\frac{3}{2},1+\frac{1}{p}))\). \(\Vert v(t) \Vert _{H^{1}(\mathbb{R})}\) is bounded if \(b=a(k+1)\). Also \(\Vert v(t) \Vert _{H^{2}(\mathbb{R})}\) is bounded if \(b=\frac{ak}{2}\). Theorem 1.2 improves the result of Theorem 5.1 in [19]. Theorem 1.3 implies that wave-breaking for a solution v occurs if its slope is unbounded. This result improves Theorem 3.1 in [18] and Theorem 5.6 in [19]. From Theorems 1.4, 1.5, and 1.6, we deduce that λ, β, a, b, and k are related to global existence and infinite propagation speed of the solutions. Parts of results in [9, 10, 13, 18, 19] are extended.

2 Proof of Theorem 1.1

We prove Theorem 1.1 in following five steps.

Step 1. Let \(v^{0} =0\). Let \((v^{i}) _{i\in \mathbb{N}} \in C(\mathbb{R}^{+};B_{p,r}^{\infty }) \) be smooth and satisfy

$$ \textstyle\begin{cases} (\partial _{t}+(a(v^{i})^{k}+\beta )\partial _{x}) v^{i+1}=G , \\ v^{i+1}(0,x)=v^{i+1}_{0}=S_{i+1}v_{0}, \end{cases} $$

and suppose

$$\begin{aligned} G ={}&P_{1}(D)\biggl[\frac{b}{k+1} \bigl(v^{i}\bigr)^{k+1}+\frac{3ak-b}{2} \bigl(v^{i}\bigr)^{k-1}\bigl(v ^{i} \bigr)_{x}^{2}\biggr] \\ &{} +P_{2}(D)\biggl[\frac{(k-1)(ak-b)}{2} \bigl(v^{i} \bigr)^{k-2}\bigl(v^{i}\bigr)_{x}^{3}- \lambda v^{i} \biggr]. \end{aligned}$$

We see \(S_{i+1}v_{0} \in B_{p,r}^{\infty }\). Then the solution \(v^{i}\in C(\mathbb{R}^{+};B_{p,r}^{\infty }) \) in (2.1) is global for all \(i\in \mathbb{N}\) by Lemma 2.5 in [13].

Step 2. It is derived from Lemma 2.4 in [13] that

$$\begin{aligned} \bigl\Vert v^{i+1} \bigr\Vert _{B_{p,r}^{s}}\leq{}& e^{C_{1}\int _{0}^{t} \Vert (v^{i}(\tau ))^{k} \Vert _{B_{p,r}^{s}} \,d\tau } \\ &{} \times \biggl[ \Vert v_{0} \Vert _{B_{p,r}^{s}}+ \int _{0}^{t} e^{-C_{1} \int _{0}^{\tau } \Vert (v^{i}(\xi ))^{k} \Vert _{B_{p,r}^{s}} \,d \xi } \bigl\Vert G(\tau ,\cdot ) \bigr\Vert _{B_{p,r}^{s}} \,d\tau \biggr]. \end{aligned}$$

The notation \(a\lesssim b\) means \(a\leq Cb\) for a certain positive constant C. We acquire the estimates

$$\begin{aligned} \bigl\Vert G(t,x) \bigr\Vert _{B_{p,r}^{s}} \lesssim \bigl( \bigl\Vert v^{i} \bigr\Vert _{B_{p,r}^{s}}+ 1 \bigr)^{k} \bigl\Vert v^{i} \bigr\Vert _{B_{p,r}^{s}} . \end{aligned}$$

That is,

$$\begin{aligned} \bigl\Vert v^{i+1} \bigr\Vert _{B_{p,r}^{s}} \leq{}& C_{2}\cdot e^{C_{2} \int _{0}^{t} ( \Vert v^{i}(\tau ) \Vert _{B_{p,r}^{s}}+1)^{k} \,d \tau } \biggl[ \Vert v_{0} \Vert _{B_{p,r}^{s}} \\ &{}+ \int _{0}^{t}e^{-C_{2} \int _{0}^{\tau }( \Vert v^{i}(\xi ) \Vert _{B_{p,r}^{s}}+1)^{k} \,d\xi }\bigl( \bigl\Vert v^{i} \bigr\Vert _{B_{p,r}^{s}}+ 1\bigr)^{k} \bigl\Vert v^{i} \bigr\Vert _{B_{p,r}^{s}} \,d \tau \biggr]. \end{aligned}$$

One may find certain \(T>0\) which satisfies \(2kC_{2}^{k+1}(1+ \Vert v_{0} \Vert _{B_{p,r}^{s}} )^{k}T<1\) and

$$ \bigl( 1+ \bigl\Vert v^{i}(t) \bigr\Vert _{B_{p,r}^{s}} \bigr)^{k}\leq \frac{C_{2} ^{k}( 1+ \Vert v_{0} \Vert _{B_{p,r}^{s}} )^{k}}{1-2kC_{2}^{k+1}( 1+ \Vert v_{0} \Vert _{B_{p,r}^{s}} )^{k}t}. $$

Further, we deduce

$$\begin{aligned} \bigl( 1+ \bigl\Vert v^{i+1}(t) \bigr\Vert _{B_{p,r}^{s}} \bigr)^{k}\leq \frac{C_{2} ^{k}( 1+ \Vert v_{0} \Vert _{B_{p,r}^{s}} )^{k}}{1-2kC_{2}^{k+1}( 1+ \Vert v_{0} \Vert _{B_{p,r}^{s}} )^{k}t}, \end{aligned}$$

which implies that \((v^{i})_{i\in \mathbb{N}}\) is uniformly bounded in \(E_{p,r}^{s}(T)\).

Step 3. Let \(m, n\in \mathbb{N}\). From (2.1), we deduce that

$$\begin{aligned} &\bigl(\partial _{t}+\bigl(a\bigl(v^{m+n} \bigr)^{k}+\beta \bigr)\partial _{x}\bigr) \bigl(v^{m+n+1}-v^{m+1}\bigr) \\ &\quad{}=-a\bigl(\bigl(v ^{m+n}\bigr)^{k}-\bigl(v^{m} \bigr)^{k}\bigr)\partial _{x}v^{m+1} \\ &\qquad{} +P_{1}(D)\biggl[\frac{b}{k+1}\bigl(\bigl(v^{m+n} \bigr)^{k+1}-\bigl(v^{m}\bigr)^{k+1}\bigr) \biggr] \\ &\qquad{} +P_{1}(D)\biggl[\frac{3ak-b}{2}\bigl(\bigl(v^{m+n} \bigr)^{k-1}\bigl(v^{m+n}\bigr)^{2}_{x}- \bigl(v ^{m}\bigr)^{k-1}\bigl(v^{m} \bigr)_{x}^{2}\bigr)\biggr] \\ &\qquad{} +P_{2}(D)\biggl[\frac{(k-1)(ak-b )}{2} \bigl(\bigl(v^{m+n} \bigr)^{k-2} \bigl(v^{m+n}\bigr)_{x} ^{3}-\bigl(v^{m }\bigr)^{k-2} \bigl(v^{m}\bigr)_{x}^{3}\bigr)\biggr] \\ &\qquad{} +P_{2}(D)\bigl[-\lambda \bigl(v^{m+n}-v^{m} \bigr)\bigr]. \end{aligned}$$

Using Lemma 2.4 in [13] yields

$$\begin{aligned} & \bigl\Vert v^{m+n+1}-v^{m+1} \bigr\Vert _{B_{p,r}^{s-1}} \\ &\quad\leq e^{C \int _{0}^{t} \Vert v^{m+n} \Vert _{B_{p,r}^{s}} ^{k} \,d \tau }\biggl[ \bigl\Vert v_{0}^{m+n+1}-v_{0}^{m+1} \bigr\Vert _{B_{p,r} ^{s-1}}+C\times \int _{0}^{t} e^{-C \int _{0}^{\tau } \Vert v^{m+n} \Vert _{B_{p,r}^{s}}^{k} \,d\xi } \\ &\qquad{}\times \bigl( \bigl\Vert v^{m+n}-v^{m} \bigr\Vert _{B_{p,r}^{s-1}}\bigl( \bigl\Vert v^{m} \bigr\Vert _{B_{p,r}^{s}}+ \bigl\Vert v^{m+n} \bigr\Vert _{B_{p,r}^{s}}+ \bigl\Vert v^{m+1} \bigr\Vert _{B_{p,r}^{s}}+1\bigr)^{k }\bigr) \,d \tau \biggr]. \end{aligned}$$

We note that the initial values satisfy

$$\begin{aligned} v_{0}^{m+n+1}-v_{0}^{m+1}=\sum _{q=m+1}^{m+n}\Delta _{q} v_{0}. \end{aligned}$$

One may find a constant \(C_{T_{1}}\) independent of m to satisfy

$$\begin{aligned} \bigl\Vert v^{m+n+1}-v^{m+1} \bigr\Vert _{L^{\infty }([0,T];B_{p,r}^{s-1})} \leq C_{T_{1}}2^{-m}. \end{aligned}$$

We obtain the desired results.

Step 4. Following the discussions in Step 4 in Sect. 3.1 in [13], one derives that \(v \in E_{p,r}^{s}(T) \), which is continuous.

Step 5. (Proof of the uniqueness). Suppose \(1\leq r, p \leq \infty , s>\operatorname{max}(\frac{3}{2}, 1+\frac{1}{p} )\). Assume \(v^{1} \) and \(v^{2} \) satisfy (1.7) with \(v_{0}^{1}, v_{0}^{2} \in B_{p,r}^{s} \), \(v^{1},v^{2}\in L^{\infty }([0,T];B_{p,r}^{s}) \cap C([0,T];B_{p,r}^{s-1})\). We write \(v^{12}=v^{1}-v^{2}\). Then

$$\begin{aligned} v^{12}\in L^{\infty }\bigl([0,T];B_{p,r}^{s} \bigr) \cap C\bigl([0,T];B_{p,r}^{s-1}\bigr), \end{aligned}$$

which results in

$$ \textstyle\begin{cases} \partial _{t}v^{12}+(a(v^{1})^{k}+\beta )\partial _{x}v^{12}=-a((v^{ 1})^{k}-(v ^{ 2})^{k})\partial _{x}v^{2}+G_{1}, \\ v^{12}(0,x)=v_{0}^{12}=v_{0}^{1}-v_{0}^{2}, \end{cases} $$


$$\begin{aligned} G_{1} ={}&P_{1}(D)\biggl[\frac{b}{k+1}\bigl( \bigl(v^{1}\bigr)^{k+1}-\bigl(v^{2} \bigr)^{k+1}\bigr)\biggr] \\ &{} +P_{1}(D)\biggl[\frac{3ak-b}{2}\bigl(\bigl(v^{1} \bigr)^{k-1}\bigl(v^{1}\bigr)^{2}_{x}- \bigl(v^{2}\bigr)^{k-1}\bigl(v ^{2} \bigr)_{x}^{2}\bigr)\biggr] \\ &{}+P_{2}(D)\biggl[\frac{(k-1)(ak-b)}{2} \bigl(\bigl(v^{1} \bigr)^{k-2} \bigl(v^{1}\bigr)_{x}^{3}- \bigl(v ^{2 }\bigr)^{k-2}\bigl(v^{2} \bigr)_{x}^{3}\bigr)-\lambda v^{12}\biggr]. \end{aligned}$$

Using Lemma 2.4 in [13], we derive the estimates

$$\begin{aligned} &e^{-C\int _{0}^{t} \Vert v^{1} \Vert _{B_{p,r}^{s }}^{k} \,d \tau } \bigl\Vert v^{12} \bigr\Vert _{B_{p,r}^{s-1}} \\ &\quad \leq \bigl\Vert v^{12} _{0} \bigr\Vert _{B_{p,r}^{s-1}} \\ &\qquad{}+ C \int _{0}^{t} e^{-C\int _{0}^{\tau } \Vert v^{1} \Vert _{B_{p,r} ^{s }} ^{k} \,d\xi } \bigl\Vert v^{12} \bigr\Vert _{B_{p,r}^{s-1}} \bigl( \bigl\Vert v^{1} \bigr\Vert _{B_{p,r}^{s}}+ \bigl\Vert v^{2} \bigr\Vert _{B_{p,r}^{s}}+1\bigr)^{k } \,d \tau , \end{aligned}$$

which finishes the proof of the uniqueness.

Remark 2.1

Suppose \(b=a(k+1), 1\leq r, p \leq \infty \), \(v_{0} \in B_{p,r}^{s} (\mathbb{R})\ (s>\operatorname{max}(1+\frac{1}{p}, \frac{3}{2} ))\), \(t\in [0,T]\). Then, the solution v to (1.1) satisfies

$$\begin{aligned} \bigl\Vert v(t) \bigr\Vert _{H^{1}}\leq \Vert v_{0} \Vert _{H^{1}}. \end{aligned}$$

3 Proofs of Theorems 1.2, 1.3, 1.4, and 1.5

3.1 Proof of Theorem 1.2

Taking advantage of the operator \(\Delta _{q} \) to (1.7) yields

$$\begin{aligned} \bigl(\partial _{t}+\bigl(av^{k}+\beta \bigr)\partial _{x}\bigr)\Delta _{q} v=a \bigl[v^{k},\Delta _{q}\bigr]\partial _{x}v+ \Delta _{q} G_{2}(t,x), \end{aligned}$$


$$\begin{aligned} G_{2}(t,x)={}&P_{1}(D)\biggl[\frac{b}{k+1}v^{k+1}+ \frac{3ak-b}{2}v^{k-1}v _{x}^{2} \biggr] \\ &{} +P_{2}(D)\biggl[\frac{(k-1)(ak-b)}{2} v^{k-2}v_{x}^{3}- \lambda v \biggr]. \end{aligned}$$

Applying Lemma 2.3 in [13] gives rise to the estimates

$$\begin{aligned} \bigl\Vert a \bigl[v^{k},\Delta _{q}\bigr]\partial _{x}v \bigr\Vert _{B_{p,r}^{s }} \lesssim \Vert v_{x} \Vert _{L^{\infty }}^{k} \Vert v \Vert _{B_{p,r}^{s }} \end{aligned}$$


$$\begin{aligned} \bigl\Vert G_{2}(t,x) \bigr\Vert _{B_{p,r}^{s }} \lesssim \bigl( \Vert v _{x} \Vert _{L^{\infty }}^{k} +1 \bigr) \Vert v \Vert _{B_{p,r}^{s }}. \end{aligned}$$

We derive that

$$\begin{aligned} \bigl\Vert v(t) \bigr\Vert _{B_{p,r}^{s }} \lesssim \Vert v_{0} \Vert _{B_{p,r}^{s }} + \int _{0}^{t} \bigl(1+ \bigl\Vert v_{x}(\tau ) \bigr\Vert _{L^{\infty }}\bigr)^{k} \bigl\Vert v(\tau ) \bigr\Vert _{B_{p,r} ^{s }} \,d\tau. \end{aligned}$$

That is,

$$\begin{aligned} \bigl\Vert v(t) \bigr\Vert _{B_{p,r}^{s }} \lesssim \Vert v_{0} \Vert _{B_{p,r}^{s }} e^{\int _{0}^{t} (1+ \Vert v_{x}(\tau ) \Vert _{L^{\infty }})^{k} \,d\tau }. \end{aligned}$$

Letting \(t\in [0,T^{\ast } ], T^{\ast } <\infty \) and

$$\begin{aligned} \int _{0}^{t} \bigl(1+ \bigl\Vert v_{x}(\tau ) \bigr\Vert _{L^{\infty }}\bigr)^{k} \,d \tau < \infty , \end{aligned}$$

we see that \(\Vert v(T^{\ast }) \Vert _{B_{p,r}^{s }}\) is bounded by using (3.2). It yields a contradiction, ending the proof.

From Remark 2.1, we obtain a blow-up result.

Remark 3.1

If assumption \(b=a(k+1)\) is added into Theorem 1.2, then condition in (1.8) is changed into

$$\begin{aligned} \int _{0}^{t} \bigl(1+ \Vert v_{x} \Vert _{L^{\infty }}\bigr)^{2} \,d\tau = \infty. \end{aligned}$$

3.2 Proof of Theorem 1.3

We only need to prove Theorem 1.3 with \(s=2\) by density argument. Take \(b=a(k+1)\). It is deduced from (1.1) that

$$\begin{aligned} \frac{1}{2}\frac{ d}{ dt} \int _{\mathbb{R}}\bigl(v^{2}+v_{x}^{2} \bigr) \,dx+ \int _{\mathbb{R}} \lambda \bigl(v^{ 2}+ v_{x}^{2}\bigr) \,dx=0, \end{aligned}$$

which results in

$$\begin{aligned} \frac{1}{2}\frac{ d}{ dt} \int _{\mathbb{R}}\bigl(v^{2}+v_{x}^{2} \bigr) \,dx\leq 0. \end{aligned}$$

A direct calculation shows that

$$\begin{aligned} & \frac{1}{2}\frac{ d}{ dt} \int _{\mathbb{R}}\bigl(v_{x}^{2}+v^{2}_{xx} \bigr) \,dx \\ &\quad=a(k+2) \int _{\mathbb{R}}v^{k}v_{x}v_{xx} \,dx - \int _{\mathbb{R}} \lambda \bigl(v_{x}^{ 2} + v _{xx}^{2}\bigr) \,dx \\ &\qquad{} - \int _{\mathbb{R}}\bigl[a(k+1) v^{k-1}v_{x}v_{xx}^{2}+av^{k}v _{xxx}v_{xx} \bigr]\,dx. \end{aligned}$$

Let \(T<\infty \) and \(v_{x}(t,x)\geq -M\) for a certain \(M>0\). We come to the estimate

$$\begin{aligned} \bigl\Vert v(t) \bigr\Vert _{H^{2}}\leq \Vert v_{0} \Vert _{H^{2}}e ^{(1+M+ \Vert v_{0} \Vert _{H^{1}})^{k}t}, \quad\text{for all } t\in [0,T], \end{aligned}$$

which yields a contradiction.

3.3 Proof of Theorem 1.4

We take \(n =v-v_{xx}\). The first equation in (1.1) is written in the form

$$\begin{aligned} n_{t}+\beta n_{x}+\lambda n +bv^{k-1}v_{x}n +av^{k} n_{x}=0. \end{aligned}$$

We see \(b=a(k+1)\) in Theorem 1.4. Multiplying (3.7) by n and applying (3.6) gives rise to

$$\begin{aligned} \frac{1}{2}\frac{ d}{ dt} \int _{\mathbb{R}}n^{2} \,dx+\lambda \int _{\mathbb{R}}n^{2} \,dx\lesssim \frac{ \vert a \vert (k+2)}{4} \Vert v_{0} \Vert _{H^{1}}^{k-1} \Vert n \Vert _{L^{2}}^{3}. \end{aligned}$$

Taking \(\lambda _{1}=2\lambda \) and \(M_{1}=\frac{|a|(k+2)}{2} \Vert v_{0} \Vert _{H^{1}}^{k-1} \), we have

$$\begin{aligned} \frac{ d}{ dt} \Vert n \Vert _{L^{2}}^{2}+ \lambda _{1} \Vert n \Vert _{L^{2}}^{2}\leq M_{1}\bigl( \Vert n \Vert _{L^{2}}^{2} \bigr)^{ \frac{3}{2}}. \end{aligned}$$

It follows that \(\Vert n \Vert _{L^{2}}\leq e^{-\frac{1}{2} \lambda _{1} t}(\frac{1}{ \Vert n_{0} \Vert _{L^{2}}}-\frac{M _{1}}{ \lambda _{1} })^{-1}\) if \(\Vert n_{0} \Vert _{L^{2}}<\frac{ \lambda _{1}}{M_{1}}\). Then

$$\begin{aligned} \Vert v_{x} \Vert _{L^{\infty }}\leq \Vert n \Vert _{L ^{2}}\leq C_{2}(T). \end{aligned}$$

Using Theorem 1.3, we end the proof.

3.4 Proof of Theorem 1.5

We investigate problem

$$ \textstyle\begin{cases} \frac{d}{dt}p(t,x)=av^{k}(t,p(t,x))+\beta , \\ p(0,x)=x, \end{cases} $$

where \((t,x)\in (0,T)\times \mathbb{R}\).

Lemma 3.1


Let \(v\in C([0,T];H^{s}(\mathbb{R})) \cap C^{1}([0,T];H^{s-1}(\mathbb{R}))\ (s\geq 2)\), \((t,x)\in [0,T] \times \mathbb{R}\). It follows that \(p\in C^{1}([0,T]\times \mathbb{R}, \mathbb{R})\) to (3.8) is unique and

$$\begin{aligned} p_{x}(t,x)=e^{\int _{0}^{t}akv^{k-1}v_{x}(\tau ,p(\tau ,x)) \,d\tau }. \end{aligned}$$

Lemma 3.2

Let \(v_{0}\in H^{s}(\mathbb{R})\ (s\geq 2)\), \((t,x)\in [0,T]\times \mathbb{R}\). Then

$$\begin{aligned} n(t,p ) (p_{x})^{\frac{b}{ak}}(t,x)=n_{0} e^{-\lambda t}. \end{aligned}$$

Moreover, \(\Vert n \Vert _{L^{\frac{ak}{b}}}=e^{-\lambda t} \Vert n_{0} \Vert _{L^{\frac{ak}{b}}}\). If \(b=\frac{ak}{2}\), it holds that

$$\begin{aligned} \Vert n \Vert _{L^{2}}=e^{-\lambda t} \Vert n_{0} \Vert _{L^{2}}. \end{aligned}$$


From (3.10), we acquire that

$$\begin{aligned} \frac{d}{dt}\bigl[n(t,p ) (p_{x})^{\frac{b}{ak}} \bigr]=-\lambda n (p_{x})^{ \frac{b}{ak}}. \end{aligned}$$

That is,

$$\begin{aligned} n(t,p ) (p_{x})^{\frac{b}{ak}}=e^{-\lambda t}n_{0}(x). \end{aligned}$$

A direct computation gives rise to

$$\begin{aligned} \bigl\Vert e^{-\lambda t}n_{0}(x) \bigr\Vert _{L^{\frac{ak}{b}}}= \Vert n \Vert _{L^{\frac{ak}{b}}}. \end{aligned}$$

We note \(b=\frac{ak}{2}\). Thus we get (3.11). □

Proof of Theorem 1.5

Multiplying (3.7) by \(ne^{2\lambda t}\), we come to

$$\begin{aligned} \frac{d}{dt}\biggl(e^{2\lambda t} \int _{\mathbb{R}}n^{2} \,dx\biggr)=(ak-2b)e^{2 \lambda t} \int _{\mathbb{R}}n^{2}v^{k-1}v_{x} \,dx. \end{aligned}$$

We derive that

$$\begin{aligned} \frac{d}{dt}\biggl(e^{2\lambda t} \int _{\mathbb{R}}n^{2} \,dx\biggr)\leq \frac{ \vert ak-2b \vert }{2^{k}}e^{-k\lambda t}\biggl[e^{2\lambda t} \int _{\mathbb{R}}n ^{2} \,dx\biggr]^{\frac{k+2}{2}}. \end{aligned}$$

Let \(h(t)=e^{2\lambda t}\int _{\mathbb{R}}n^{2} \,dx\). Bearing in mind that \(n_{0}(x)\neq 0\), \(x\in \mathbb{R}\) and (3.10), one deduces that \(h(t) \) is positive. Then

$$\begin{aligned} \frac{d}{dt}\bigl[h(t)\bigr]^{-\frac{k}{2}}\geq - \frac{k}{2} \frac{ \vert ak-2b \vert }{2^{k}}e^{-k\lambda t}. \end{aligned}$$

Using the assumption \(n_{0}(x)\neq 0, b\neq \frac{ak}{2} \), \(\Vert n_{0} \Vert _{L^{2}}< ({\frac{2^{k+1}\lambda }{|ak-2b|}})^{ \frac{1}{k}}\), we have \([h(0)]^{-\frac{k}{2}} - \frac{|ak-2b|}{2^{k+1} \lambda }> 0\). We obtain the inequality

$$\begin{aligned} \biggl(e^{2\lambda t} \int _{\mathbb{R}} n^{2} \,dx\biggr)^{\frac{k}{2}}\leq \biggl[ \Vert n_{0} \Vert _{L^{2}}^{-k}- \frac{ \vert ak-2b \vert }{2^{k+1}\lambda }\biggr]^{-1}. \end{aligned}$$

Consequently, we have the estimate

$$\begin{aligned} \Vert v_{x} \Vert _{L^{\infty }} \leq \Vert n \Vert _{L ^{2}}\leq e^{-\lambda t}\biggl[ \Vert n_{0} \Vert _{L^{2}}^{-k}-\frac{ \vert ak-2b \vert }{2^{k+1} \lambda } \biggr]^{-\frac{1}{k}}. \end{aligned}$$

Applying Theorem 1.3, we complete the proof. □

We give a global existence result.

Lemma 3.3

Let \(b=a(k+1)\) or \(b=\frac{ak}{2}\), \(v_{0} \in H^{s}(\mathbb{R})\ (s\geq 2)\). Assume \(n_{0}=v_{0}-v_{0,xx}\) does not change sign. It holds that a solution \(v(t,x)\) to problem (1.1) exists globally.


One may assume \(n_{0}(x)> 0\). We use Lemma 3.2 to derive that \(n > 0\). Thus

$$\begin{aligned} v(t,x)= \int _{\mathbb{R}}\frac{1}{2}e^{-|x-\xi |} n(t,\xi ) \,d\xi \geq 0. \end{aligned}$$

That is,

$$\begin{aligned} v(t,x)=\frac{1}{2}e^{-x} \int _{-\infty }^{x} e^{\xi }n(t,\xi ) \,d \xi +\frac{1}{2}e^{x} \int _{x}^{\infty } e^{-\xi }n(t,\xi ) \,d \xi. \end{aligned}$$

We conclude that

$$\begin{aligned} v_{x}(t,x)=-\frac{1}{2}e^{-x} \int _{-\infty }^{x} e^{\xi }n(t,\xi ) \,d \xi +\frac{1}{2}e^{x} \int _{x}^{\infty } e^{-\xi }n(t,\xi ) \,d \xi. \end{aligned}$$

Hence \(|v_{x} |\leq v \).

Applying \(b=a(k+1)\) and recalling Remark 2.1, we derive

$$\begin{aligned} \vert v_{x} \vert \leq \vert v \vert \lesssim \bigl\Vert v(t) \bigr\Vert _{H^{1}}\lesssim \Vert v_{0} \Vert _{H^{1}}. \end{aligned}$$

Taking advantage of \(b=\frac{ak}{2}\) and using Lemma 3.2 results in

$$\begin{aligned} \vert v_{x} \vert \leq \vert v \vert \lesssim \Vert n \Vert _{L^{2}}\lesssim \Vert n_{0} \Vert _{L^{2}}. \end{aligned}$$

Combining (3.18) or (3.19) with Theorem 1.2, we obtain the desired results. □

4 Proof of Theorem 1.6

Note that \(a>0\). Using \(\operatorname{supp} v_{0}(x)\subset [a_{0},b_{0}]\), we derive that \(\operatorname{supp} v_{0}(x)\subset [p(t,a_{0}),p(t,b_{0})]\). Applying Lemma 3.2 yields that \(\operatorname{supp} n(t,x)\subset [p(t,a _{0}),p(t,b_{0})]\), \(t\in [0,T]\).


$$\begin{aligned} L_{+}(t)= \int _{p(t,a_{0})}^{p(t,b_{0})} e^{\xi }n(t,\xi ) \,d \xi ,\qquad L_{-}(t)= \int _{p(t,a_{0})}^{p(t,b_{0})} e^{-\xi }n(t,\xi ) \,d \xi. \end{aligned}$$

From (3.16) and (4.1), we have

$$\begin{aligned} v(t,x)={}&\frac{1}{2}e^{-x}\biggl( \int _{-\infty }^{p(t,a_{0})}+ \int _{p(t,a _{0})}^{p(t,b_{0})}+ \int _{p(t,b_{0})}^{x}\biggr) e^{\xi }n(t,\xi ) \,d \xi \\ &{} +\frac{1}{2}e^{x} \int _{x}^{\infty } e^{-\xi }n(t,\xi ) \,d\xi \\ ={}&\frac{1}{2}e^{-x}L_{+}(t),\quad x>p(t,b_{0}). \end{aligned}$$

We derive \(v =\frac{1}{2}e^{x}L_{-}(t)\) if \(x< p(t,a_{0})\). Combining (3.17) with (4.2) gives rise to

$$\begin{aligned} v =-v_{x} =v_{xx} = \frac{1}{2}e^{-x}L_{+}(t),\quad x>p(t,b_{0}) \end{aligned}$$


$$\begin{aligned} v =v_{x} =v_{xx} = \frac{1}{2}e^{x}L_{-}(t), \quad x< p(t,a_{0}). \end{aligned}$$

An application of (4.1) leads to the identity

$$\begin{aligned} L_{+}(0)= \int _{a_{0}}^{b_{0}} e^{\xi }n_{0}( \xi ) \,d\xi =0. \end{aligned}$$

A direct calculation shows

$$\begin{aligned} \frac{ d }{ dt} L_{+}(t) ={}& \int _{-\infty }^{\infty } e^{\xi }n_{t}(t, \xi ) \,d\xi \\ ={}&{-} \int _{-\infty }^{\infty } e^{\xi }(\lambda -\beta )n\,d\xi + \int _{-\infty }^{\infty } e^{\xi } \frac{b}{k+1} v^{k+1} \,d\xi \\ &{} +\frac{3ak-b}{2} \int _{-\infty }^{\infty } e^{\xi }v_{x}^{2} v^{k-1} \,d \xi + \frac{ (k-1)(ak-b)}{2} \int _{-\infty }^{\infty } e^{\xi }v_{x}^{3} v ^{k-2} \,d\xi. \end{aligned}$$

If \(b=ak \) and k is a positive odd number, we obtain

$$\begin{aligned} \frac{ d }{ dt} L_{+}(t)+(\lambda -\beta )L_{+}(t) >0, \end{aligned}$$

which is equivalent to the inequality

$$\begin{aligned} \frac{ d [L_{+}(t)e^{(\lambda -\beta )t}]}{ dt}>0. \end{aligned}$$

Hence \(L_{+}(t)>0\), \(t\in [0,T)\).

Similarly, we have

$$\begin{aligned} \frac{ d [-L_{-}(t)e^{(\lambda +\beta )t}]}{ dt}>0. \end{aligned}$$

Thus, \(L_{-}(t)<0\), \(t\in [0,T)\).

If \(k=1, 0< b<3a\), we derive that (4.8) and (4.9) still hold true.

We give the estimates for curve \(p(t,b_{0})\). Using the assumption \(k=1, b=2a \) and (3.4) yields

$$\begin{aligned} \Vert v \Vert _{L^{\infty }}\leq \Vert v \Vert _{H^{1}} \leq e^{-\lambda t} \Vert v_{0} \Vert _{H^{1}}. \end{aligned}$$

Taking \(x=b_{0}\) in (3.8) and integrating (3.8) on \([0,t]\), we come to the estimate

$$\begin{aligned} p(t,b_{0})&=b_{0}+ \int _{0}^{t} a v(\tau ,p ) \,d \tau +\beta t \\ &\leq \frac{1}{\lambda }C_{5}+b_{0}+\beta t. \end{aligned}$$

We conclude from (4.2) that

$$\begin{aligned} L_{+}(t)=2e^{p(t,b_{0})}v\bigl(t,p(t,b_{0}) \bigr)\leq C_{3}e^{(\beta -\lambda )t}. \end{aligned}$$

Similar to the derivation in (4.11), we have

$$\begin{aligned} p(t,a_{0})&=a_{0}+ \int _{0}^{t} av(\tau ,p ) \,d \tau +\beta t \\ &\geq -\frac{1}{\lambda }C_{5}+a_{0}+\beta t, \end{aligned}$$

which, combining with (4.4), implies

$$\begin{aligned} \bigl\vert L_{-}(t) \bigr\vert \leq C_{4}e^{-(\beta +\lambda )t}. \end{aligned}$$

If \(k=1, b=\frac{a}{2} \), it is deduced from (3.11) that \(\Vert v \Vert _{L^{\infty }} \leq e^{-\lambda t} \Vert v_{0} \Vert _{H^{2}}\). Similarly, we establish (4.12) and (4.14).

Remark 4.1

If \(\operatorname{supp} v_{0}(x)\subset [a_{0},b_{0}] \) in (1.1), then \(n =(1-\partial _{x}^{2})v(t,x)\) satisfies \(\operatorname{supp} n\subset [p(t,a_{0}),p(t,b_{0})]\). Indeed, v does not have compact support. Also \(v(t,x) \) is positive if \(x\rightarrow \infty \) and \(v(t,x) \) is negative if \(x\rightarrow -\infty \).


  1. Constantin, A.: Global existence of solutions and breaking waves for a shallow water equation: a geometric approach. Ann. Inst. Fourier (Grenoble) 50, 321–362 (2000)

    Article  MathSciNet  Google Scholar 

  2. Fu, Y., Qu, C.Z.: Well-posedness and wave breaking of the degenerate Novikov equation. J. Differ. Equ. 263, 4634–4657 (2017)

    Article  MathSciNet  Google Scholar 

  3. Gao, Y., Li, L., Liu, J.G.: A dispersive regularization for the modified Camassa–Holm equation. SIAM J. Math. Anal. 50, 2807–2838 (2018)

    Article  MathSciNet  Google Scholar 

  4. Guo, Z.G.: On an integrable Camassa–Holm type equation with cubic nonlinearity. Nonlinear Anal., Real World Appl. 34, 225–232 (2017)

    Article  MathSciNet  Google Scholar 

  5. Guo, Z.G., Li, X.G., Yu, C.: Some properties of solutions to the Camassa–Holm-type equation with higher order nonlinearities. J. Nonlinear Sci. 28, 1901–1914 (2018)

    Article  MathSciNet  Google Scholar 

  6. Himonas, A., Holliman, C., Kenig, C.: Construction of 2-peakon solutions and ill-posedness for the Novikov equation. SIAM J. Math. Anal. 50, 2968–3006 (2018)

    Article  MathSciNet  Google Scholar 

  7. Himonas, A., Mantzavinos, D.: An ab-family equations with peakon traveling waves. Proc. Am. Math. Soc. 144, 3797–3811 (2016)

    Article  MathSciNet  Google Scholar 

  8. Himonas, A., Thompson, R.: Persistence properties and unique continuation for a generalized Camassa–Holm equation. J. Math. Phys. 55, 091503 (2014)

    Article  MathSciNet  Google Scholar 

  9. Hu, Q.Y., Qiao, Z.J.: Global existence and propagation speed for a generalized Camassa–Holm model with both dissipation and dispersion (2015) arXiv:1511.03325v1

  10. Lai, S.Y., Wu, Y.H.: A model containing both the Camassa–Holm and Degasperis–Procesi equations. J. Math. Anal. Appl. 374, 458–469 (2011)

    Article  MathSciNet  Google Scholar 

  11. Li, M.G., Zhang, Q.T.: Generic regularity of conservative solutions to Camassa–Holm type equations. SIAM J. Math. Anal. 49, 2920–2949 (2017)

    Article  MathSciNet  Google Scholar 

  12. Mi, Y.S., Liu, Y., Guo, B.L., Luo, T.: The Cauchy problem for a generalized Camassa–Holm equation. J. Differ. Equ. 266, 6739–6770 (2019)

    Article  MathSciNet  Google Scholar 

  13. Ming, S., Lai, S.Y., Su, Y.Q.: The Cauchy problem of a weakly dissipative shallow water equation. Appl. Anal. 98, 1387–1402 (2019)

    Article  MathSciNet  Google Scholar 

  14. Molinet, L.: A Liouville property with application to asymptotic stability for the Camassa–Holm equation. Arch. Ration. Mech. Anal. 230, 185–230 (2018)

    Article  MathSciNet  Google Scholar 

  15. Novruzov, E., Hagverdiyev, A.: On the behavior of the solution of the dissipative Camassa–Holm equation with the arbitrary dispersion coefficient. J. Differ. Equ. 257, 4525–4541 (2014)

    Article  MathSciNet  Google Scholar 

  16. Tu, X.Y., Liu, Y., Mu, C.L.: Existence and uniqueness of the global conservative weak solutions to the rotation Camassa–Holm equation. J. Differ. Equ. 266, 4864–4900 (2019)

    Article  MathSciNet  Google Scholar 

  17. Wu, X.: On the finite time singularities for a class of Degasperis–Procesi equations. Nonlinear Anal., Real World Appl. 44, 1–17 (2018)

    Article  MathSciNet  Google Scholar 

  18. Yan, W., Li, Y.S., Zhang, Y.M.: Global existence and blow-up phenomena for the weakly dissipative Novikov equation. Nonlinear Anal. 75, 2464–2473 (2012)

    Article  MathSciNet  Google Scholar 

  19. Zhang, L., Liu, B.: On the Cauchy problem for a class of shallow water wave equations with \((k+1)\)-order nonlinearities. J. Math. Anal. Appl. 445, 151–185 (2017)

    Article  MathSciNet  Google Scholar 

  20. Zhou, Y., Chen, H.P.: Wave breaking and propagation speed for the Camassa–Holm equation with \(k\neq 0\). Nonlinear Anal., Real World Appl. 12, 1875–1882 (2011)

    Article  MathSciNet  Google Scholar 

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We are grateful to the anonymous referees for a number of valuable comments and suggestions.

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The project is supported by Science Foundation of North University of China (No. 2017030, No. 13011920) and the National Natural Science Foundation of P.R. China (No. 11471263).

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Su, Y., Lai, S. & Ming, S. The dynamic properties of solutions for a nonlinear shallow water equation. Bound Value Probl 2019, 163 (2019).

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