The Cauchy problem for the generalized Degasperis-Procesi equation

In this paper, we investigate the Cauchy problem for the generalized Degasperis-Procesi equation in a Besov space. Firstly, we prove that the generalized Degasperis-Procesi equation is locally well posed in B_{p,r}^s with s>1+\frac{1}{p} (or s\ge 1+\frac{1}{p} if r=1 with p\in [1,+\mathrm{\infty })). Secondly, we prove that the generalized Degasperis-Procesi equation possesses the peaked solitary wave which is the weak solution to the generalized Degasperis-Procesi equation. Thirdly, we prove that the data-to-solution map for the generalized Degasperis-Procesi equation is not uniformly continuous in B_{2,\mathrm{\infty }}^{3/2}. Fourthly, we prove that the data-to-solution map for the generalized Degasperis-Procesi equation is not uniformly continuous in H^s(\mathbf{R}) with . Finally, we give a blow-up criterion.

MSC:35G25, 35L05.

In this paper, we consider the Cauchy problem for the following generalized Degasperis-Procesi equation:

where Q\in \mathbf{R} is a constant. When k=1 and Q=\frac{3}{2}, (1.1) reduces to the Degasperis-Procesi equation

Equation (1.3) possesses the Lax pair and bi-Hamiltonian structures and infinite many conservation laws [1]. The Degasperis-Procesi equation [2] possesses peaked solitons which are stable [3] and shock peakons of the form u(x,t)=-\frac{1}{t+k}sign(x)e^{-|x|}, k>0. The Degasperis-Procesi equation possesses the global weak solution and blow-up structure [4–6]. Constantin and Lannes studied the relevance between the Camassa-Holm equation and the Degasperis-Procesi equation [7]. The Degasperis-Procesi equation possesses the infinite propagation speed [8]. The Degasperis-Procesi equation possesses multi-peakon solutions [9] and multisoliton [10]. Himonas and his co-authors [11, 12] proved that the data-to-solution for the Camassa-Holm equation, the Degasperis-Procesi equation is not uniformly continuous in H^s(\mathbf{R}) with s>3/2, respectively. Himonas et al. proved the non-uniform continuity in H^1 of the solution map of the CH equation [13]. Recently, Gui and Liu [14] studied the Cauchy problem for the Degasperis-Procesi equation in Besov spaces. Yan et al. [15] studied the Cauchy problem for the Novikov equation in Besov spaces.

Let P(D)=-{\partial }_x{(1-{\partial }_x^2)}^{-1} and p(x)=\frac{1}{2}{e}^{-|x|}, x\in \mathbf{R}. By using the identity {(1-{\partial }_x^2)}^{-1}f=p\ast f for f\in L^2, we can rewrite (1.1)-(1.2) as follows:

In this paper, motivated by [14], we study the Cauchy problem for (1.4) in Besov spaces. Firstly, we use the standard iterative method to prove that the generalized Degasperis-Procesi equation is locally well posed in B_{p,r}^s with s>1+\frac{1}{p} (or s\ge 1+\frac{1}{p} if r=1 with p\in [1,+\mathrm{\infty })). Secondly, we prove that the generalized Degasperis-Procesi equation possesses the peaked solitary wave which is the weak solution to the generalized Degasperis-Procesi equation. Thirdly, we prove that the data-to-solution map for the generalized Degasperis-Procesi equation is not uniformly continuous in B_{2,\mathrm{\infty }}^{3/2}. Fourthly, we prove that the data-to-solution map for the generalized Degasperis-Procesi equation is not uniformly continuous in H^s(\mathbf{R}) with . Finally, we give a blow-up criterion.

Notice that the structure of (1.4) is more complicated than that of the Degasperis-Procesi equation. Thus, to prove that the sequence of smooth solutions {(u^{(n)})}_{n\in \mathbf{N}} is uniformly bounded in C([0,T];B_{p,r}^s)\cap C^1([0,T];B_{p,r}^{s-1}) with s>1+\frac{1}{p} (or s\ge 1+\frac{1}{p} if r=1 with p\in [1,+\mathrm{\infty })), we choose that

In proving Theorem 1.5, we explain why we choose (1.6). It is worthy of pointing out that we use Fatou’s lemma and the upper limit as well as Gronwall’s inequality to prove that {(u^{(n)})}_{n\in \mathbf{N}} is a Cauchy sequence in C([0,T];B_{p,r}^{s-1}) with s>1+\frac{1}{p} (or s\ge 1+\frac{1}{p} if r=1 with p\in [1,+\mathrm{\infty })).

To introduce the main results, we define

The main results of this paper are as follows.

Theorem 1.1 Let u_0(x)\in B_{p,r}^s with s>1+\frac{1}{p} (or s\ge 1+\frac{1}{p} if r=1 with p\in [1,+\mathrm{\infty })). Problem (1.4)-(1.5) is locally well posed. Moreover,

A function u:[0,T)\times \mathbf{R} is called a weak solution to (1.1) (or (1.4)) if u belongs to L_{\mathrm{loc}}^{\mathrm{\infty }}([0,T);H^1) and satisfies the following identity:

where p(x)=\frac{1}{2}{e}^{-|x|}, for any smooth test function \varphi (x,t)\in C_c^{\mathrm{\infty }}([0,T)\times \mathbf{R}). If u is a weak solution on [0,T) for every T>0, then it is called a global weak solution.

Theorem 1.2 When Q=\frac{k(k+2)}{k+1} in (1.4), u_c(x,t)=c^{1/k}{e}^{-|x-ct|} with c>0 is a weak solution of (1.4) in the sense of (1.8).

Theorem 1.3 When Q=\frac{k(k+2)}{k+1} in (1.4), the data-to-solution map for the generalized Degasperis-Procesi equation is not uniformly continuous in B_{2,\mathrm{\infty }}^{3/2}. More precisely, there exists a global solution u\in L^{\mathrm{\infty }}({\mathbf{R}}^{+};B_{2,\mathrm{\infty }}^{3/2}) to the Cauchy problem for (1.4) such that for any T>0 and ϵ>0, there exists a solution v\in L^{\mathrm{\infty }}(0,T;B_{2,\mathrm{\infty }}^{3/2}) with

Theorem 1.4 When Q=\frac{k(k+2)}{k+1} in (1.4), the data-to-solution map for the generalized Degasperis-Procesi equation is not uniformly continuous in H^s(\mathbf{R}) with .

Theorem 1.5 Assume that T^{\star } is the maximal time of existence of the solution to problem (1.4)-(1.5). If , then

Moreover, T^{\star }\ge \frac{1}{Ck{\parallel u_0\parallel }_{B_{p,r}^s}^k}.

The remainder of this paper is organized as follows. In Section 2, we give some preliminaries. In Section 3, we prove Theorem 1.1. In Section 4, we prove Theorem 1.2. In Section 5, we prove Theorem 1.3. In Section 6, we prove Theorem 1.4. In Section 7, we prove Theorem 1.5.

In this section, we give Lemmas 2.1-2.4. The proof of Lemmas 2.1-2.4 can be seen in [16–21].

Lemma 2.1 (Littlewood-Paley decomposition)

Let B=\{\xi \in {\mathbf{R}}^n,|\xi |\le \frac{4}{3}\} and C=\{\xi \in {\mathbf{R}}^n,\frac{3}{4}\le |\xi |\le \frac{8}{3}\}. There exists a couple of smooth radial functions (\chi ,\varphi )\in (C_c^{\mathrm{\infty }}(B),C_c^{\mathrm{\infty }}(C)) such that

and

and

Then, for u\in {\mathcal{S}}^{\prime }(\mathbf{R}), the nonhomogeneous dyadic blocks are defined as follows:

Thus we obtain

and the low frequency cut-off S_q is defined by

as well as

where C is a positive constant independent of q.

Definition (Besov spaces)

Let s\in \mathbf{R} and 1\le p\le +\mathrm{\infty }. The nonhomogeneous Besov space B_{p,r}^s({\mathbf{R}}^n) is defined by

In particular, if s=\mathrm{\infty }, then B_{p,r}^s={\bigcap }_{s\in \mathbf{R}}{B}_{p,r}^s.

Lemma 2.2 Let s\in \mathbf{R}, 1\le p,r,p_j,r_j\le \mathrm{\infty }, j=1,2, then:

1. (1)

   B_{p,r}^s is a Banach space and is continuously embedded in {\mathcal{S}}^{\prime }({\mathbf{R}}^n).

2. (2)

   B_{p_1,r_1}^{s_1}\hookrightarrow B_{p_2,r_2}^{s_2}, if p_1\le p_2 and r_1\le r_2 and s_2=s_1-n(\frac{1}{p_1}-\frac{1}{p_2})
3. (3)

   \mathrm{\forall }s>0, B_{p,r}^s\cap L^{\mathrm{\infty }} is a Banach algebra. B_{p,r}^s is a Banach algebra iff B_{p,r}^s\hookrightarrow L^{\mathrm{\infty }} and iff s>\frac{1}{p} or (s\ge \frac{1}{p} and r=1). (4)

4. (i)

   For s>0,

   {\parallel fg\parallel }_{B_{p,r}^s}\le C({\parallel f\parallel }_{B_{p,r}^s}{\parallel g\parallel }_{L^{\mathrm{\infty }}}+{\parallel f\parallel }_{L^{\mathrm{\infty }}}{\parallel g\parallel }_{B_{p,r}^s}),\mathrm{\forall }f,g\in B_{p,r}^s\cap L^{\mathrm{\infty }}.
5. (ii)

   (s_2\ge \frac{1}{p} if r=1) and s_1+s_2>0,

   {\parallel fg\parallel }_{B_{p,r}^{s_1}}\le C{\parallel f\parallel }_{B_{p,r}^{s_1}}{\parallel g\parallel }_{B_{p,r}^{s_2}},\mathrm{\forall }f\in B_{p,r}^{s_1},g\in B_{p,r}^{s_2}.
6. (5)

   \mathrm{\forall }\theta \in [0,1] and s=\theta s_1+(1-\theta )s_2,

   {\parallel f\parallel }_{B_{p,r}^s}\le C{\parallel f\parallel }_{B_{p,r}^{s_1}}^{\theta }{\parallel f\parallel }_{B_{p,r}^{s_2}}^{1-\theta },\mathrm{\forall }f\in B_{p,r}^{s_1}\cap B_{p,r}^{s_2}.
7. (6)

   If {(u_n)}_{n\in \mathbf{N}} is bounded in B_{p,r}^s and u_n\to u in {\mathcal{S}}^{\prime }({\mathbf{R}}^n), then u\in B_{p,r}^s and

   {\parallel u\parallel }_{B_{p,r}^s}\le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}{\parallel u_n\parallel }_{B_{p,r}^s}.
8. (7)

   Let m\in \mathbf{R} and Ψ be an S^m-multiplier. Then the operator \mathrm{\Psi }(D) is continuous from B_{p,r}^s into B_{p,r}^{s-m}. In particular, -{\partial }_x{(1-{\partial }_x^2)}^{-1} is continuous from B_{p,r}^s into B_{p,r}^{s-1}.

Lemma 2.3 (A priori estimates in Besov spaces)

Let 1\le p,r\le \mathrm{\infty } and s>-min\{\frac{1}{p},1-\frac{1}{p}\}. Assume that f_0\in B_{p,r}^s, F\in L^1(0,T;B_{p,r}^s) and that {\partial }_{x}v belongs to L^1(0,T;B_{p,r}^{s-1}) if s>1+\frac{1}{p} or to L^1(0,T;B_{p,r}^{1/p}\cap L^{\mathrm{\infty }}) otherwise. If f\in L^{\mathrm{\infty }}(0,T;B_{p,r}^s)\cap C([0,T];{\mathcal{S}}^{\prime }(\mathbf{R})) solves the following 1-D linear transport equation:

then there exists a constant C depending only on s, p, r such that the following statements hold:

1. (1)

   If r=1 or s\ne 1+\frac{1}{p}, then

   {\parallel f\parallel }_{B_{p,r}^s}\le {\parallel f_0\parallel }_{B_{p,r}^s}+{\int }_0^t{\parallel F(\tau )\parallel }_{B_{p,r}^s}d\tau +C{\int }_0^t{V}^{\prime }(\tau ){\parallel f(\tau )\parallel }_{B_{p,r}^s}d\tau

or

with V(t)={\int }_0^t{\parallel v_x(\tau )\parallel }_{B_{p,r}^{1/p}\cap L^{\mathrm{\infty }}}d\tau if and V(t)={\int }_0^t{\parallel v_x(\tau )\parallel }_{B_{p,r}^{s-1}}d\tau else.

1. (2)

   If s\le 1+\frac{1}{p}, f_0^{\prime }\in L^{\mathrm{\infty }} and f_x\in L^{\mathrm{\infty }}((0,T)\times \mathbf{R}) and F_x\in L^1(0,T;L^{\mathrm{\infty }}), then

   \begin{array}{c}{\parallel f(t)\parallel }_{B_{p,r}^s}+{\parallel f_x(t)\parallel }_{L^{\mathrm{\infty }}}\hfill \\ \le e^{CV(t)}({\parallel f_0\parallel }_{B_{p,r}^s}+{\parallel f_0^{\prime }\parallel }_{L^{\mathrm{\infty }}}+{\int }_0^t{e}^{-CV(\tau )}[{\parallel F(\tau )\parallel }_{B_{p,r}^s}+{\parallel F_x(\tau )\parallel }_{L^{\mathrm{\infty }}}]d\tau )\hfill \end{array}

with

1. (3)

   If f=v, then for all s>0, (1) holds true when V(t)={\int }_0^t{\parallel v_x(\tau )\parallel }_{L^{\mathrm{\infty }}}d\tau.

2. (4)

   If , then f\in C([0,T];B_{p,r}^s). If r=\mathrm{\infty }, then f\in C([0,T];B_{p,1}^{s^{\prime }}) for all .

Lemma 2.4 (Existence and uniqueness)

Let p, r, s, f_0 and F be as in the statement of Lemma  2.3. Assume that v\in L^{\rho }(0,T;B_{\mathrm{\infty },\mathrm{\infty }}^{-M}) for some \rho >1 and M>0 and v_x\in L^1(0,T;B_{p,r}^{s-1}) if s>1+\frac{1}{p} or s=1+\frac{1}{p} and r=1 and v_x\in L^1(0,T;B_{p,\mathrm{\infty }}^{1/p}\cap L^{\mathrm{\infty }}) if . Then problem (2.1)-(2.2) has a unique solution and the inequalities of Lemma  2.3 can hold true. Moreover, if , then f\in C([0,T];B_{p,r}^s).

In this section, we complete the proof of Theorem 1.1 and suppose that s>1+\frac{1}{p} (or s\ge 1+\frac{1}{p} if r=1 with p\in [1,+\mathrm{\infty })).

First step: approximate solution

By using the standard iterative process, we construct a sequence of smooth solutions {(u^{(n)})}_{n\in \mathbf{N}}\in C({\mathbf{R}}^{+};B_{p,r}^{\mathrm{\infty }}). Assume that u^{(0)}:=0, by induction we define a sequence of smooth functions {(u^{(n)})}_{n\in \mathbf{N}} by solving the following linear transport equation:

By using the fact that S_{n+1}{u}_0 belong to B_{p,r}^{\mathrm{\infty }}, from Lemma 2.4, for all n\in \mathbf{N}, we can show by induction that problem (3.1)-(3.2) has a global solution {(u^{(n)})}_{n\in \mathbf{N}}\in C({\mathbf{R}}^{+},B_{p,r}^{\mathrm{\infty }}).

Second step: uniform bounds

For s>1+\frac{1}{p} (or s\ge 1+\frac{1}{p} if r=1 with p\in [1,+\mathrm{\infty })) and n\in \mathbf{N}, we prove that

with U^n={\int }_0^t{\parallel u^{(n)}\parallel }_{B_{p,r}^s}^{k}d\tau.

From (2.3) of Lemma 2.3 and (3.1), we derive that

where

By using the S^{-1} multiplier property of P(D) and the fact B_{p,r}^s with s>1+\frac{1}{p} (or s\ge 1+\frac{1}{p} if r=1 with p\in [1,+\mathrm{\infty })) is a Banach algebra, we have

Inserting (3.6) into (3.4) yields (3.3).

Let us fix T>0 such that

and suppose that

Since U^n(t)={\int }_0^t{\parallel u^{(n)}\parallel }_{B_{p,r}^s}^{k}d\tau, by using (3.8), we have

When \tau =0 in (3.9), we have

Inserting (3.9), (3.10) into (3.3) yields

Thus, {(u^{(n)})}_{n\in \mathbf{N}} is uniformly bounded in C([0,T];B_{p,r}^s). By using (3.9) and the fact that B_{p,r}^s is a Banach algebra and (3.5) and using the S^{-1}-multiplier property of P(D), we have that

and

Consequently,

Remark Inserting (3.7) into (3.8) yields

for n\in \mathbf{N}. From (3.11) and (3.7), \mathrm{\forall }n\in {\mathbf{N}}^{+}, we have that

and

with the aid of Fatou’s lemma. We define

Thus,

for n\in {\mathbf{N}}^{+}.

Third step: convergence

We prove that {(u^{(n)})}_{n\in \mathbf{N}} is a Cauchy sequence in C([0,T];B_{p,r}^{s-1}). For (m,n)\in {\mathbf{N}}^2, we have

Combining (2.3) with (3.20), we have that

where

By using (3) and (7) of Lemma 2.2, we have that

Inserting (3.22) into (3.21) yields

From (3.2) and Lemma 2.1, we can easily obtain that

Obviously,

Inserting (3.24) and (3.25) into (3.23) leads to

We define

Inserting (3.27) into (3.26) leads to

We define

and

Combining (3.30) with (3.29), (3.28), by using Fatou’s lemma, we have that

Applying Gronwall’s inequality to (3.31) yields

for t\in [0,T]. According to the definition of \tilde{\rho }(t), we can easily obtain that

Combining (3.32) with (3.33), we have that

Hence, {(u^n)}_n is a Cauchy sequence in C([0,T];B_{p,r}^{s-1}).

Fourth step: existence in E_{p,r}^s(T)

Now we prove that u\in E_{p,r}^s(T) and satisfies (1.4)-(1.5) since {(u^n)}_{n\in \mathbf{N}} is uniformly bounded in L^{\mathrm{\infty }}(0,T;B_{p,r}^s). From (6) in Lemma 2.2, we have that u\in L^{\mathrm{\infty }}(0,T;B_{p,r}^s). From (1.4), we can easily prove that u_t\in L^{\mathrm{\infty }}(0,T;B_{p,r}^{s-1}). It is easily checked that u is indeed a solution to (1.4)-(1.5) by passing to the limit in (3.1)-(3.2).

Now we prove that u\in E_{p,r}^s(T). Since u\in B_{p,r}^s, \mathrm{\forall }ϵ>0, there exists q_0\in {\mathbf{N}}^{+} such that

From the definition of Besov spaces, we have that

By using the mean value theorem, we have that

where . Inserting (3.37) into (3.36) yields