Abstract
This study focuses on the Cauchy problem associated with the two-component peakon system featuring a cubic nonlinearity, constrained to the class \((m,n)\in C^{k}(\mathbb {R}) \cap W^{k,1}(\mathbb {R})\) with \(k\in \mathbb {N}\cup \{0\}\). This system extends the celebrated Fokas–Olver–Rosenau–Qiao equation and the following nonlocal (two-place) counterpart proposed by Lou and Qiao:
where \(m(t,x)=\left( 1-\partial _{x}^2\right) u(t,x)\). Employing an approach based on Lagrangian coordinates, we establish the local existence, uniqueness, and Lipschitz continuity of the data-to-solution map in the class \(C^k\cap W^{k,1}\). Moreover, we derive criteria for blow-up of the local solution in this class.
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1 Introduction
We consider the Cauchy problem for the following two-component peakon system with cubic nonlinearity introduced by Song, Qu and Qiao in [33]:
for \(m=m(t,x),\,u=u(t,x),\,v=v(t,x)\) and \(t,x\in \mathbb {R}\). We assume that the initial data \(u_0(x)=u(0,x)\) and \(v_0(x)=v(0,x)\) belong to the space \(C^{k+2}(\mathbb {R})\cap W^{k+2,1}(\mathbb {R})\) with \(k\in \mathbb {N}_0\), where \(\mathbb {N}_0=\mathbb {N}\cup \{0\}\). Taking \(u=v\) in (1.1), one obtains the Fokas–Olver–Rosenau–Qiao (FORQ) equation, also referred to as the modified Camassa–Holm equation, which has the form
Fokas and Fuchssteiner originally introduced this equation as an integrable variant of the modified Korteweg–de Vries (mKdV) equation, known to possess peakon solutions (see [9, Equation (7)] and [10, Equation (26f)]). Subsequently, leveraging its bi-Hamiltonian structure, Olver and Rosenau, along with Schiff, derived (1.2) as a dual counterpart to the mKdV equation, see [28] and [31]. Later, Qiao [29] further advanced the development of the FORQ equation as an approximation of the two-dimensional Euler equations, where u represents the fluid velocity, and m corresponds to its potential density. Additionally, (1.2) can be reduced to the short pulse (SP) equation,
by the scaling transformation \(x'=\frac{x}{\varepsilon }\), \(t'=\varepsilon t\), \(u'=\frac{u}{\varepsilon ^2}\) and passing to the limit \(\varepsilon \rightarrow 0\) [15]. The SP equation was proposed by Schäfer and Wayne [32], and it is useful for modeling the propagation of ultra-short light pulses in silica optics. Lastly, it is worth noting that the FORQ equation can be found in the list of equations compiled by Novikov, taking the form \((1-\partial _x^2)\partial _tu=F(u,\partial _xu,\partial _x^2,\dots )\), where F represents a quadratic or cubic differential polynomial in u and its derivatives with respect to x, see [27, Equation (32)].
The FORQ equation, along with its generalizations, has been the subject of extensive research, exploring its well-posedness and blow-up properties in several works [8, 11, 12, 15, 16, 38, 40,41,42,43]. In particular, the geometric formulation of (1.2) can be found in [15, Section 2]. Additionally, a wide range of exact solutions for the FORQ equation, including algebro-geometric, peakon, smooth, and loop-shaped solutions, have been derived and discussed in various studies [3, 4, 20, 25, 29]. Moreover, the inverse scattering method and the long-term behavior of solutions to the Cauchy problem associated with (1.2) have been explored in [4, 22].
Another intriguing reduction of (1.1) is the nonlocal (two-place) FORQ equation, originally introduced by Lou and Qiao [24, equation (26)]:
Equation (1.3) can be linked to (1.1) by setting \(v(t,x)=u(-t,-x)\), thereby allowing us to derive it following a methodology akin to that employed by Ablowitz and Musslimani in introducing various nonlocal variations of well-known integrable equations [1, 2]. Specifically, this approach can be applied to obtain the nonlocal counterpart of the nonlinear Schrödinger (NLS) equation
The works [1, 2] considered the following integrable Ablowitz–Kaup–Newell–Segur (AKNS) system:
which, in the case \(r(t,x)=\sigma \overline{q(t,-x)}\), reduces to the nonlocal NLS equation
The conventional NLS (1.4) corresponds to \(r(t,x)=\sigma \overline{q(t,x)}\). It is worth noting that (1.5) exhibits nonlocal behavior exclusively in the spatial variable x.
Returning to the nonlocal FORQ equation (1.3), we observe that it incorporates solution values from non-adjacent points, such as x and \(-x\). This unique feature allows for the description of phenomena characterized by intrinsic correlations and entanglement between events taking place at distinct locations [23]. The nonlocal FORQ equation (1.3) was initially derived as a reduction of the following system, which was introduced by Xia, Qiao, and Zhou (see [36, Equation (7)]):
with \(v(t,x)=u(-t,-x)\) and \(H(t,x)=2(u(t,x)-\partial _xu(t,x)) (u(-t,-x)+\partial _x(u(-t,-x)))\). This system satisfies the parity-time-symmetric (PT-symmetric) condition, i.e., \(H(t,x)=H(-t,-x)\).
The Cauchy problem for the system (1.1) can be written in the following nonlocal form (see [21, equations (4.1a), (4.1c)] and [26]):
with initial data
where
with
The Cauchy problem’s local well-posedness within a range of Besov spaces and its associated blow-up criteria have been thoroughly examined in previous studies, see [26] and [39] respectively. For additional references, see [35] and [37]. Moreover, exact solutions of (1.6a), including those involving multipeakons, have been successfully derived and investigated in [39]. The work [7] explores the spectral aspects of the two-component system, particularly in the context of multipeakons. Lastly, we mention that the Hamiltonian duality between (1.1) and other integrable systems has been rigorously established in [34] and [21].
In our current work, we focus on the study of the Cauchy problem (1.6) within the class of functions u and v belonging to \(C([-T,T],C^{k+2}(\mathbb {R}) \cap W^{k+2,1}(\mathbb {R}))\), where \(k\in \mathbb {N}_0\). Our approach involves revisiting the method of characteristics, as previously developed in [12] and [42], primarily for addressing the FORQ equation. This method has been successfully applied to various peakon equations, enabling one to obtain global solutions that may exhibit finite-time singularities, see [5, 6, 18, 19]. Moreover, it has proven valuable in the analysis of problems featuring nonzero asymmetric asymptotics for u(t, x) as x approaches both positive and negative infinity, as demonstrated in [14]. It is important to note that the Cauchy problem for the nonlocal FORQ equation (1.3) can exhibit distinct qualitative properties when the asymptotic behavior at positive and negative infinity differs significantly, resembling step-like patterns. This distinctive behavior arises from the non-translation invariance of (1.3), in contrast to the conventional FORQ equation (1.2). We refer to [30] for a related discussion for the nonlocal NLS equation (1.5) with step-like boundary conditions.
By utilizing the explicit representation of the solution (u, v), derived from the known initial data and the characteristics, as shown in (3.4) below, we establish the existence and uniqueness of local solutions within the space \(C^{k+2}\cap W^{k+2,1}\). We note that our chosen class of regularity, specifically for the case where \(k=0\), exhibits a lower regularity exponent than what has been previously explored in works related to the FORQ equation and its generalizations, as exemplified in [15] and [42]. One of the most challenging aspects of our analysis lies in proving uniqueness when \(k=0\). To achieve this, we must demonstrate that every solution adheres to a specific conservation law, as outlined in (3.3) below. This entails examining equation (1.1) in a weak sense, a task that is further detailed in Lemma 3.9 below.
Next, we establish the Lipschitz continuity property of the data-to-solution map for (m, n) within the space \(C^k\cap W^{k,1}\), where \(k\in \mathbb {N}_0\). To the best of our knowledge, this particular result has not been previously documented. In related works, such as [12, Section 4.1] and [43, Section 4.1], it was demonstrated that, assuming the existence of a weak solution for the FORQ equation in \(W^{2,1}\), the solution itself exhibits a Lipschitz property within the space \(W^{1,1}\). For solutions of the FORQ equation residing in \(H^s\) with \(s>\frac{5}{2}\), the data-to-solution map maintains continuity but does not possess uniform continuity, as discussed in [16]. For further insights into the continuity properties of the FORQ equation and the two-component system (1.1) within the context of \(H^s\) spaces, we refer to [17] and [21].
Finally, we establish new blow-up criteria for the solution within the space \(C^{k+2}\cap W^{k+2,1}\), where \(k\in \mathbb {N}_0\). This extends and generalizes previous findings presented in [12], which were primarily centered on compactly supported classical solutions of (1.1) with \(k\in \mathbb {N}\).
The structure of this article unfolds as follows. In Sect. 2, we lay the foundation by introducing essential notations, definitions, and relevant facts that will serve as the basis for our subsequent discussions. Section 3 is dedicated to the development of the Lagrangian approach for addressing the Cauchy problem (1.6). Within this section, we leverage Lagrangian coordinates to establish the local existence, uniqueness, and Lipschitz continuity of the data-to-solution map. A summary of these results is provided in Theorem 3.13. Finally, Sect. 4 focuses on the task of establishing blow-up criteria for the solution in \(C^{k+2}\cap W^{k+2,1}\), where \(k\in \mathbb {N}_0\).
2 Preliminaries
In this section we introduce some notations and facts to be used throughout the paper. We use the following functional spaces:
where \(k\in \mathbb {N}_0\). Also it is convenient for us to use the following notations for the Banach spaces
Note that when f belongs to either \(C^{k}(\mathbb {R})\) or \(W^{k,\infty }(\mathbb {R})\), where \(k\in \mathbb {N}\), then f is a bounded function. Throughout this text, we adopt the convention of writing \(L^1\), and similarly for other function spaces, without specifying \(\mathbb {R}\) when it does not introduce ambiguity to the reader. Moreover, we use the spaces
of k-times continuously differentiable functions \(u(t):[-T_1,T_2]\mapsto X\) with the norm
Finally, we will use the following elementary inequalities
as well as the notation \(\textrm{id}\) for the identity function.
3 Local solutions in Lagrangian coordinates
3.1 Lagrangian dynamics
We introduce the following flow map for the two-component system (1.1) (cf. [35, 39]):
It turns out that using (3.1), we are able to obtain an explicit representation for the solution (u, v)(t, x) in terms of \(y(t,\xi )\) and the known initial data. All the derivations below are formal and will be justified later. Form (1.1) we have
which imply
where \(m_0(x)=m(0,x)\), \(n_0(x)=n(0,x)\) and \(y_0(\xi )=y(0,\xi )\).
Assume that \(y(t,\xi )\rightarrow \pm \infty \) as \(\xi \rightarrow \pm \infty \) and \(y(t,\xi )\) is strictly monotone increasing in \(\xi \) for all fixed t. Taking into account that \(u(t,x)=\frac{1}{2}e^{-|x|}*m(t,x)\) and using (3.3a), we can obtain the following formula for the component u(t, x) (cf. [12, equation (8)]):
Arguing similarly for v(t, x), we obtain the following representations for the components u and v:
Then (3.4) imply that
The equations (3.4) and (3.5) lead us to consider (3.1), subject to the initial condition \(y_0(\xi )\), as a Cauchy problem for an ordinary differential equation in a Banach space. Here, the ODE vector field is characterized by the known data \(m_0(x)\) and \(n_0(x)\). To properly formulate this problem, we proceed to define
and
Observe that when \(m_0(x)\) and \(n_0(x)\) belong to the space \(L^1(\mathbb {R})\), we can establish the following uniform estimate with respect to t for the functions U, W, V, and Z:
assuming only that \(y_0(\xi )\) exhibits monotonic behavior and tends toward \(\pm \infty \) as \(\xi \rightarrow \pm \infty \). Also it is convenient for us to introduce the function
which will turn out to be bounded for \(\xi \in \mathbb {R}\), see (3.26) below.
Taking into account (3.1) and that \(\partial _t\zeta (t,\xi )=\partial _ty(t,\xi )\), we obtain the following Cauchy problem:
which is considered in the Banach space \(E_\ell \subset C^{1}(\mathbb {R})\) defined as follows (cf. [18, Section 2.2] and [12, equation (16)])
with the norm \(\Vert f\Vert _{E_\ell }=\Vert f\Vert _{C^{1}(\mathbb {R})}\). Notice that if \(\zeta (t,\cdot )\in E_\ell \) with \(\ell >0\), then \(\partial _\xi y(t,\xi )\ge \ell \) for all \(\xi \) and therefore \(y(t,\cdot )\) is strictly monotone increasing.
3.2 Local characteristic
Introduce the following operator, corresponding to (3.10):
In Proposition 3.2 below, we will prove that A is a contraction in \(C([-T,T],E_\ell )\) for a class of initial data \(y_0(\xi )\) and sufficiently small \(T>0\). To demonstrate this, we establish the following technical lemma:
Lemma 3.1
Suppose that \(m_0,n_0\in X^0\), \(y(t,\cdot )\) is strictly monotone increasing for all \(t\in [-\tilde{T},T]\) and \(\zeta \in C\left( [-\tilde{T},T], C^1(\mathbb {R})\right) \), for some \(\tilde{T},T>0\). Then we have
-
1.
\(U,W,V,Z\in C\left( [-\tilde{T},T], C^1(\mathbb {R})\right) \);
-
2.
the partial derivatives of U, W, V, Z in \(\xi \) have the form
$$\begin{aligned} \begin{aligned}&\partial _\xi \begin{pmatrix} U(t,\xi )\\ W(t,\xi ) \end{pmatrix}= \begin{pmatrix} 0&{} \partial _\xi y(t,\xi )\\ \partial _\xi y(t,\xi )&{}0 \end{pmatrix} \begin{pmatrix} U(t,\xi )\\ W(t,\xi ) \end{pmatrix} -\begin{pmatrix} 0\\ m_0(y_0(\xi ))\partial _\xi y_0(\xi ) \end{pmatrix},\\&\partial _\xi \begin{pmatrix} V(t,\xi )\\ Z(t,\xi ) \end{pmatrix}= \begin{pmatrix} 0&{} \partial _\xi y(t,\xi )\\ \partial _\xi y(t,\xi )&{}0 \end{pmatrix} \begin{pmatrix} V(t,\xi )\\ Z(t,\xi ) \end{pmatrix} -\begin{pmatrix} 0\\ n_0(y_0(\xi ))\partial _\xi y_0(\xi ) \end{pmatrix}. \end{aligned} \end{aligned}$$(3.13)
Proof
Consider the integrals
Differentiating (3.15) with respect to \(\xi \), direct calculations show (3.13) and thus we have item (2) of the lemma.
Now let us prove item (1). First, we show that \(J_1(t,\cdot ;m_0)\) is continuous. Denoting \(\tilde{m}_0(\eta )=m_0(y_0(\eta )) \partial _\eta y_0(\eta )\) and using (2.2a), we have for any \(\xi _1,\xi _2\in \mathbb {R}\), \(\xi _1\le \xi _2\) (here we drop the arguments t, \(m_0\))
Since the condition \(\xi _1\le \xi _2\) does not restrict the generality, we have that \(J_1\) is continuous in \(\xi \). Arguing similarly, we conclude that \(J_2(t,\cdot ;m_0)\) and \(J_j(t,\cdot ;n_0)\), \(j=1,2\) are continuous, which, together with the estimates \(\left| J_j(m_0)\right| \le \Vert m_0\Vert _{L^1}\) and \(\left| J_j(n_0)\right| \le \Vert n_0\Vert _{L^1}\), imply that \(J_j(t,\cdot ;m_0)\), \(J_j(t,\cdot ;n_0)\) belong to \(C(\mathbb {R})\).
Now let us prove that these functions belong to \(C\left( [-\tilde{T},T], C(\mathbb {R})\right) \). As above, we consider \(J_1(t)=J_1(t,\xi ;m_0)\) only, the other integrals can be treated similarly. For all \(t_1,t_2\in [-\tilde{T},T]\),
where we have used (2.2a). Since \(\zeta \in C\left( [-\tilde{T},T],C^1(\mathbb {R})\right) \), we have that \(J_j(t,\xi ;m_0), J_j(t,\xi ;n_0)\in C\left( [-\tilde{T},T],C(\mathbb {R})\right) \) and (3.15) implies that U, W, V, Z also belong to this Banach space. Finally, using (3.13) and that \(\partial _\xi y\in C\left( [-\tilde{T},T],C(\mathbb {R})\right) \), we arrive at item 1 of the lemma.\(\square \)
Now we are at the position to prove the contraction of the operator A defined by (3.12).
Proposition 3.2
Consider \(m_0,n_0\in X^0\). Assume that \((y_0-\textrm{id})\in E_c\) for some \(c>0\). Take any \(0<\ell <c\). Then for all
the operator A defined in (3.12) is a contraction in \(C([-T,T],E_\ell )\):
for all \(\zeta _j\in C([-T,T],E_\ell )\), \(j=1,2\) and some \(0<\alpha <1\), \(\alpha =\alpha (T)\).
Proof
Firstly, note that a function \(y_0\) meeting the conditions of the proposition does indeed exist. This can be achieved by choosing \(y_0(\xi )=c\xi \), where c is a positive constant.
Let us prove that \(A(\zeta )\in C\left( [-T,T],E_\ell \right) \) for any \(\zeta \in C([-T,T],E_\ell )\), where \(E_\ell \) is defined in (3.11). Differentiating (3.12) in \(\xi \) and applying (3.13), we obtain
Combining (3.12), (3.18) and item (1) in Lemma 3.1, we conclude that \(A(\zeta )\in C\left( [-T,T],C^1(\mathbb {R})\right) \).
It remains to prove that \(\partial _\xi A(\zeta )(t,\xi )\ge \ell -1\) for all \((t,\xi )\in [-T,T]\times \mathbb {R}\). From (3.18) and (3.8), we have the following inequality:
for all \((t,\xi )\in [-T,T]\times \mathbb {R}\). Taking T as in (3.16), we conclude that \(\partial _\xi A(\zeta )(t,\xi )\ge \ell -1\).
Now let us prove (3.17). Let \(U_j\), \(W_j\), \(V_j\) and \(Z_j\) \(j=1,2\), denote U, W, V and Z, respectively, with \(\zeta _j(t,\xi )=y_j(t,\xi )-\xi \) instead of \(\zeta (t,\xi )=y(t,\xi )-\xi \). Using that \(y(t,\cdot )\) is strictly monotone increasing, the inequality (2.2a) and that \(||a|-|b||\le |a-b|\) for all \(a,b\in \mathbb {R}\), we have
for all \((t,\xi )\in [-T,T]\times \mathbb {R}\). Arguing similarly, we obtain
Using Item (1) of Lemma 3.1, (3.19) and (2.2c), it implies from (3.12) that (we drop the arguments \(t,\xi \) for \(A(\zeta _j)\), \(j=1,2\))
To estimate \(|\partial _\xi A(\zeta _1)-\partial _\xi A(\zeta _2)|\), we use (3.18) together with (3.19), which imply
Combining (3.20) and (3.21) with T satisfying (3.16), we arrive at (3.17).\(\square \)
Using Proposition 3.2, we can easily prove that there exists a unique local solution of the Cauchy problem (3.10) in the Banach space \(C([-T,T],E_\ell )\), see Proposition 3.4 below. However, to establish the decay rate of the solution \(\zeta (t,\cdot )\), we will need the following lemma.
Lemma 3.3
Suppose that \(m_0,n_0\in X^0\), \(\zeta \in C^1\left( [-T,T], C(\mathbb {R})\right) \) for some \(T>0\) and \((y_0-\textrm{id})\in E_\ell \), \(\ell >0\). Then \(U,W,V,Z\in C\left( [-T,T],L^1(\mathbb {R})\right) \).
Proof
In view of (3.15), it is enough to show that the integrals \(J_j(m_0)\) and \(J_j(n_0)\), \(j=1,2\), defined in (3.14) belong to \(C\left( [-T,T],L^1(\mathbb {R})\right) \). We give a proof for \(J_1(t,\xi ;m_0)\), the other integrals can be treated similarly.
Changing the order of integration and using that \(\Vert \zeta (t,\cdot )\Vert _{C(\mathbb {R})} =\Vert y(t,\cdot )-(\cdot )\Vert _{C(\mathbb {R})}\) is finite for all \(t\in [-T,T]\), we have
Now let us establish the continuity of the map \(t\mapsto \Vert J_1(t)\Vert _{L^1}\). Changing the order of integration as in (3.22), we have for any \(t_1,t_2\in [-T,T]\)
The integral \(I_1\) can be estimated by using the mean value theorem and taking into account that \(\partial _ty=\partial _t\zeta \), see (3.9), as follows:
for some \(t^*\) between \(t_1\) and \(t_2\). In a similar manner we can estimate \(I_2\) and thus eventually conclude that \(J_j(m_0),J_j(n_0)\in C\left( [-T,T],L^1(\mathbb {R})\right) \), \(j=1,2\).\(\square \)
Now we can show that there exists a unique local solution \(\zeta (t,\xi )\) of the Cauchy problem (3.10), which has additional regularity and decay rate for a class of initial data \(m_0,n_0\).
Proposition 3.4
(Existence and uniqueness of the local characteristics) Suppose that \(m_0\), \(n_0\in X^k\), \((y_0(\xi )-\xi )\in X^{k+1}\) for some \(k\in \mathbb {N}_0\) and \((y_0-\textrm{id})\in E_c\), \(c>0\). Then for any \(0<\ell <c\) and T satisfying (3.16) there exists a unique \(\zeta \in C([-T,T],E_\ell )\) such that
which is a unique local solution of the Cauchy problem (3.10) in the Banach space \(C([-T,T],E_\ell )\). Moreover, the solution \(\zeta (t,\xi )\) has the following regularity and decay properties:
Finally, \(\zeta (t,\xi )\) satisfies the following size estimates:
with
and some \(C_j=C_j(\Vert \partial _\xi ^j y_0\Vert _C,\Vert m_0\Vert _{X^{j}}, \Vert n_0\Vert _{X^{j}})>0\).
Proof
The existence and uniqueness of the fixed point of the operator A in the Banach space \(C([-T,T], E_\ell )\) follow from Proposition 3.2 and the contraction mapping theorem.
Let us prove that \(\zeta \in C([-T,T], C^{k+1}(\mathbb {R}))\) by induction. We already know (see (3.11)) that \(\zeta \in C\left( [-T,T], C^{1}(\mathbb {R})\right) \) and thus the base case of the induction is established. Suppose that \(\zeta \in C\left( [-T,T], C^{j}(\mathbb {R})\right) \), for some \(j=1,\dots ,k\). To show that \(\zeta \in C\left( [-T,T], C^{j+1}(\mathbb {R})\right) \), we notice that (3.18) implies
where the right-hand side does not depend on \(\partial _\xi \zeta (t,\xi )\). Therefore, (3.27) and item (2) of Lemma 3.1 imply that
where \(\mathcal {P}_r\), \(r=1,\dots ,4\), are polynomials which depend on U, W, V, Z, \(\{\partial _\xi ^i\zeta \}_{i=0}^{j}\), \(\{\partial _\xi ^im_0\}_{i=0}^{j}\), \(\{\partial _\xi ^in_0\}_{i=0}^{j}\) and \(\{\partial _\xi ^i(y_0(\xi )-\xi )\}_{i=1}^{j+1}\). Using item (1) of Lemma 3.1 as well as the induction hypothesis, we conclude that \(\zeta \in C\left( [-T,T],C^{j+1}(\mathbb {R})\right) \), which completes the induction step. Therefore, we have established that \(\zeta \in C\left( [-T,T],C^{k+1}(\mathbb {R})\right) \). Arguing similarly for \(\partial _t\zeta \) and \(\partial _\xi \partial _t\zeta \), we conclude that \(\partial _t\zeta \in C\left( [-T,T],C^{k+1}(\mathbb {R})\right) \) and therefore \(\zeta \in C^1\left( [-T,T],C^{k+1}(\mathbb {R})\right) \).
We also establish that \(\zeta \in C\left( [-T,T],W^{k+1,1}(\mathbb {R})\right) \) through an inductive approach. Lemma 3.3, (3.24), (3.27), and (3.8) imply that \(\zeta \in C\left( [-T,T], W^{1,1}(\mathbb {R})\right) \). The induction step can be proved by using (3.28) and applying again Lemma 3.3, item (1) of Lemma 3.1 and the induction hypothesis, that is, \(\zeta \in C\left( [-T,T], W^{j,1}(\mathbb {R})\right) \), \(j=1,\dots k\). Reasoning in the similar manner, we can prove that \(\partial _t\zeta \in C\left( [-T,T], W^{k+1,1}(\mathbb {R})\right) \).
Finally, inequalities (3.26a) and (3.26b) with \(j=0\) follow from (3.8), (3.24) and (3.27), while (3.26b) for \(j=1,\dots ,k\) follows from (3.28).\(\square \)
Remark 3.5
Notice that the time \(T>0\) in (3.16) depends on the initial data \(m_0,n_0\) and it cannot be extended by choosing certain specific initial value \(y_0\) of the Cauchy problem (3.10).
Finally, let us prove the continuous dependence of \(\zeta \) on \(m_0,n_0\), which will be used in Sect. 3.3 for proving the Lipschitz continuity properties of the solution (m, n), see Corollary 3.12 below.
Proposition 3.6
(Lipschitz continuity of \(\zeta \) on \((m_0,n_0)\)) Fix any two constants \(0<R_0\le R\). Suppose that \(m_{0,j},n_{0,j}\in X^{k}\), \(j=1,2\), for some \(k\in \mathbb {N}_0\), are such that
Also assume that \((y_0-\textrm{id})\in X^{k+1}\) and \((y_0-\textrm{id})\in E_c\) for some \(c>0\). Consider the corresponding characteristics (see Proposition 3.4)
where \(\hat{U}_j\), \(\hat{W}_j\), \(\hat{V}_j\) and \(\hat{Z}_j\) are defined by (3.6), (3.7) with \(\zeta _j=y_j-\xi \), \(m_{0,j}\) and \(n_{0,j}\) instead of y, \(m_0\) and \(n_0\) respectively, \(j=1,2\). Then we have the following Lipschitz property of \(\zeta _j\) for a sufficiently small \(T>0\) (here we drop the arguments of functions for simplicity):
for some \(C_1=C_1(T,\Vert y_0(\cdot )-(\cdot )\Vert _{C^1},R_0)>0\) and \(C_{2,r}= C_{2,r}(T,\Vert y_0(\cdot )-(\cdot )\Vert _{C^{r+1}},R)>0\).
Proof
Introduce the following integrals (cf. (3.14)):
Then we have
where \(j=1,2\).
First, we prove (3.30a). Observe that (here \(\hat{J}_{1,j}(m_{0,j}) =\hat{J}_{1,j}(t,\xi ;m_{0,j})\), \(j=1,2\))
where in the second inequality we have used (2.2a). Arguing similarly for \(|\hat{J}_{2,1}(m_{0,1})-\hat{J}_{2,2}(m_{0,2})|\) and \(|\hat{J}_{i,1}(n_{0,1})-\hat{J}_{i,2}(n_{0,2})|\), \(i=1,2\), we conclude from (3.31) (we drop the arguments for simplicity)
Then combining (3.29) and (2.2c), (3.8) for \(U_j\), \(W_j\), \(V_j\), \(Z_j\), \(j=1,2\), we arrive at (here \(C_T=C([-T,T],C(\mathbb {R}))\))
for some \(\tilde{C}= \tilde{C}(\Vert m_{0,j}\Vert _{L^1},\Vert n_{0,j}\Vert _{L^1})\). Inequality (3.33) together with (3.32) implies (3.30a), for a sufficiently small \(T>0\), with the norm \(\Vert \cdot \Vert _{C([-T,T],C(\mathbb {R}))}\) utilized on the left-hand side.
To obtain (3.30a) for the norm \(\Vert \cdot \Vert _{C([-T,T],L^1(\mathbb {R}))}\), we should estimate the \(L^1\)-norms of the differences on the left-hand side of (3.32). Notice that
where (here we change the order of integration and use the notation \(\tilde{M}_{0,1}(\eta )=|m_{0,1}(y_0(\eta ))\partial _\eta y_0(\eta )|\))
Recalling that \(y_j(t,\xi )=\zeta _j(t,\xi )-\xi \) and using (2.2b), we obtain
The integral \(I_4\) can be estimated as follows:
Arguing similarly for \(\Vert \hat{J}_{2,1}(m_{0,1}) -\hat{J}_{2,2}(m_{0,2})\Vert _{L^1}\) and \(\Vert \hat{J}_{i,1}(n_{0,1}) -\hat{J}_{i,2}(n_{0,2})\Vert _{L^1}\), \(i=1,2\), we obtain
Using (3.29) and (2.2c), (3.8) for \(U_j\), \(W_j\), \(V_j\), \(Z_j\), \(j=1,2\), we arrive at (cf. (3.33))
for some \(\tilde{C}= \tilde{C}(\Vert m_{0,j}\Vert _{L^1},\Vert n_{0,j}\Vert _{L^1})\). The latter inequality together with (3.34) and (3.26a) implies (3.30a) with the norm \(\Vert \cdot \Vert _{C([-T,T],L^1(\mathbb {R}))}\). Recalling that \(X^0=C\cap L^1\), we conclude that (3.30a) is proved.
Then observing that (cf. (3.27))
and arguing as in the proof of (3.30a), we obtain (3.30b) for \(r=0\). Finally, successively differentiating (3.35) with respect to \(\xi \) and applying item (2) in Lemma 3.1 for \(\hat{U}_j\), \(\hat{W}_j\), \(\hat{V}_j\) and \(\hat{Z}_j\), we arrive at (3.30b) for \(r=1,\dots ,k\).\(\square \)
3.3 Local well-posedness
In this section we will prove that the pair (u, v) defined by (3.4) is a unique local solution of the Cauchy problem (1.6) in \(X^{k+2}\). Moreover, we will show that the data-to-solution map from the initial data \((m_0,n_0)\) to (m, n) is Lipschitz continuous (see Theorem 3.13 below).
We start with establishing the regularity properties of u and v defined by (3.4) as well as their decay rate for the large |x|.
Proposition 3.7
(Regularity and decay of u and v) Assume that \(m_0,n_0\in X^{k}\), \(k\in \mathbb {N}_0\) and consider the local characteristic \(y(t,\xi )=\zeta (t,\xi )-\xi \) obtained in Proposition 3.4. Then the functions u(t, x) and v(t, x), defined by (3.4a) and (3.4b) respectively, satisfy the following regularity and decay conditions (here \(T>0\) is the same as in Proposition 3.4):
Proof
Introduce the following integrals, cf. (3.14) (recall that \(\partial _\xi y(t,\xi )\ge \ell \) for all \((t,\xi )\in [-T,T]\times \mathbb {R}\) for some \(\ell >0\), which implies that \(y(t,\cdot )\) is a bijection from \(\mathbb {R}\) to \(\mathbb {R}\)):
where \(\xi =[y(t)]^{-1}(x)\) is such that \(y(t,\xi )=x\). In these notations, we have (see (3.4))
Moreover, since
we conclude that
Let us show that \(\tilde{J}_j(t,x;m_0)\) and \(\tilde{J}_j(t,x;n_0)\), \(j=1,2\), belong to the space \(C\left( [-T,T],C^{k+1}(\mathbb {R})\right) \). We give a detailed proof for \(\tilde{J}_1(t,x;m_0)\), the other integrals can be treated similarly. Observing that \(|\tilde{J}_1(t,x;m_0)|\le \Vert m_0\Vert _{L^1}\) and
where we have used (2.2a), we conclude that \(\tilde{J}_1(t,x;m_0)\in L^\infty ([-T,T],C(\mathbb {R}))\).
Then take any \(t_1,t_2\in [-T,T]\) and let \(\xi _j=\xi _j(x)=[y(t_j)]^{-1}(x)\), \(j=1,2\). Since \(y(t_1,\xi _1)=y(t_2,\xi _2)=x\) and using the mean value theorem, we obtain
for certain values of \(t^*\) and \(\xi ^*\) lying between \(t_1\) and \(t_2\), and \(\xi _1\) and \(\xi _2\), respectively. We have the following inequality (here the arguments \(x,m_0\) of \(\tilde{J}_1(t,x;m_0)\) are dropped):
Recalling that \(x=y(t_1,\xi _1)\) and \(\partial _\xi y(t,\xi )\ge \ell \) and applying the mean value theorem, the integral \(I_5\) can be estimated as follows (here we denote \(\tilde{M}_0=\Vert m_0\Vert _C\Vert \partial _\xi y_0\Vert _C\)):
Taking into account that \(e^{\xi ^*-\xi _1}\le e^{|\xi _1-\xi _2|}\), we have from (3.41) and (3.43) that
The integral \(I_6\) can be estimated as follows (as above, we denote \(\tilde{M}_0=\Vert m_0\Vert _C\Vert \partial _\xi y_0\Vert _C\)):
for some \(t^*\) between \(t_1\) and \(t_2\) (here we have used that \(|[y(t_2)]^{-1}(x)-x|\le \Vert \zeta (t_2,\cdot )\Vert _C\)).
Combining (3.42), (3.44) and (3.45), we conclude that \(\tilde{J}_1(t,x;m_0)\in C([-T,T],C(\mathbb {R}))\). Then (3.39) implies that \(\tilde{J}_1(t,x;m_0)\in C\left( [-T,T],C^1(\mathbb {R})\right) \). Using (3.25) and successively differentiating (3.39) with respect to x, we conclude that \(\tilde{J}_1(t,x;m_0)\in C\left( [-T,T],C^{k+1}(\mathbb {R})\right) \). Arguing similarly, we can prove that \(\tilde{J}_2(m_0),\tilde{J}_j(n_0)\in C\left( [-T,T],C^{k+1}(\mathbb {R})\right) \), \(j=1,2\), which, together with (3.38) and (3.40), imply that \(u,v\in C\left( [-T,T], C^{k+2}(\mathbb {R})\right) \).
Now let us prove that \(\tilde{J}_j(m_0),\tilde{J}_j(n_0)\), \(j=1,2\), belong to the space \(C\left( [-T,T], W^{k+1,1}(\mathbb {R})\right) \). As above, we give a detailed proof for \(\tilde{J}_1(m_0)\), all other integrals can be analyzed in a similar manner. By Fubini’s Theorem,
which implies that \(\tilde{J}_1\in L^\infty \left( [-T,T],L^1(\mathbb {R})\right) \).
For any \(t_1,t_2\in [-T,T]\) we have (recall the notation \(\xi _j=[y(t_j)]^{-1}(x)\), \(j=1,2\); here we drop the arguments \(x,m_0\) of \(\tilde{J}_1(t,x;m_0)\) for simplicity):
Applying the Fubini’s Theorem and the mean value theorem, we obtain
for some \(t^*\) between \(t_1\) and \(t_2\). Using similar arguments for \(I_8\), we arrive at the following inequality:
for some \(t^*\) between \(t_1\) and \(t_2\). Combining (3.46), (3.47) and (3.48), we have that \(\tilde{J}_1(m_0)\in C\left( [-T,T],L^1(\mathbb {R})\right) \). In view of (3.39), we eventually conclude that \(\tilde{J}_1(m_0)\in C\left( [-T,T],W^{k+1,1}(\mathbb {R})\right) \). Arguing similarly for \(\tilde{J}_2(m_0)\) and \(\tilde{J}_j(n_0)\), \(j=1,2\), the equations (3.38) and (3.40) imply that \(u,v\in C\left( [-T,T],W^{k+2,1}(\mathbb {R})\right) \).
Finally, let us show that \(\partial _tu,\partial _tv\in C([-T,T],X^{k+1})\). Introducing
we obtain that (cf. (3.40))
Condition (3.25) implies that \(\partial _ty(t,\xi )m_0(y_0(\xi )) \partial _\xi y_0(\xi )\in C\left( [-T,T],X^{k}\right) \). Therefore, using
and applying a similar line of reasoning as we did for \(\tilde{J}_1(m_0)\) earlier, we conclude that \(\check{J}_j(m_0),\check{J}_j(n_0)\) belong to \(C\left( [-T,T],X^{k+1}\right) \). which, together with (3.49), implies that \(\partial _tu,\partial _tv\in C\left( [-T,T],X^{k+1}\right) \).\(\square \)
We are now in a position to demonstrate that u and v, as defined in (3.4) through the characteristics y, constitute a solution to the Cauchy problem (1.6).
Proposition 3.8
(Local existence) Suppose that \(m_0,n_0\in X^{k}\), \(k\in \mathbb {N}_0\). Consider the local characteristics \(y(t,\xi )=\zeta (t,\xi )-\xi \) obtained in Proposition 3.4 and define the functions u(t, x) and v(t, x) by (3.4a) and (3.4b), respectively. Then u and v satisfy (3.36) and the pair (u, v) is a solution of the Cauchy problem (1.6).
Proof
Given that \(y(0,x)=y_0(\xi )\), it becomes evident that the functions u and v defined by (3.4) fulfill the initial conditions \(u(0,x)=u_0(x)\) and \(v(0,x)=v_0(x)\). Subsequently, we establish (1.6a) for u, with analogous reasoning being applicable to v. Using (3.3a), which follows from (3.6), we have
where \(\delta (x)\) is the Dirac delta function. Utilizing (3.50), we obtain for any \(\phi (x)\in \mathcal {S}(\mathbb {R})\),
which implies (1.6a) for u.\(\square \)
To establish the uniqueness of the weak solution for the FORQ equation in \(W^{2,1}(\mathbb {R})\), one can typically rely on its representation as a first-order equation, as discussed in [43, Section 4] and [13, Section 4.1]. However, the two-component system (1.1) (and the nonlocal FORQ equation) cannot be converted into a first-order equation, as is evident from terms like \((u\partial _x^2z-(\partial _x^2w)v)\) in (1.7). Therefore, in order to establish the uniqueness of (1.6), alternative arguments need to be employed. Specifically, by adhering to the Lagrangian approach, we demonstrate that any solution of (1.6) within the class (3.36), must take the form (3.4).
Lemma 3.9
Suppose that (u, v) is a local solution of the Cauchy problem (1.6), which satisfy (3.36) for some \(T>0\) and \(k\in \mathbb {N}_0\). Consider \((y_0-\textrm{id})\in E_c\) for some \(c>0\). Then
-
1.
there exists a unique solution \(y(t,\xi )\) of (3.1) subject to the initial data \(y_0(\xi )\), such that \((y-\textrm{id}_\xi )\in C^1([-T,T]\times \mathbb {R})\), where \(\textrm{id}_\xi (t,\xi )=\xi \), and \(\partial _\xi y(t,\xi )>\ell \) for all \((t,\xi )\in [-T,T]\times \mathbb {R}\), where \(0<\ell <c\), and \(T>0\) is sufficiently small;
-
2.
the equalities (3.3) hold.
Proof
The vector field \((\partial _xu-u)(\partial _xv+v)\) in (3.1) is bounded in x and is of class \(C^1\) in x and t. Therefore, the classical Cauchy theorem for ODEs implies that there exists a unique solution \(y(t,\xi )\), \((y-\textrm{id}_\xi )\in C^1\left( [-T,T]\times \mathbb {R}\right) \), with \(\textrm{id}_\xi (t,\xi )=\xi \), of the Cauchy problem for (3.1) with the initial data \(y_0(\xi )\). Since \(\partial _\xi y_0(\xi )\ge c\) for all \(\xi \in \mathbb {R}\) and \(\partial _\xi y\in C([-T,T]\times \mathbb {R})\), we have item (1) of the lemma.
Let us demonstrate item (2). Consider \(k\in \mathbb {N}\). We focus on the classical solution to the Cauchy problem for (1.1). Given this, we can directly establish the validity of (3.2) and, thus, (3.3).
In the case \(k=0\) we have that \(m(t,\cdot )\), \(n(t,\cdot )\in X^{0}\) and \(\partial _t\partial _xu(t,\cdot )\), \(\partial _t\partial _xv(t,\cdot )\in X^{0}\) for all fixed \(t\in [-T,T]\). Therefore, (1.1) can be considered for all fixed t as an equality of functionals acting on \(W^{1,\infty }(\mathbb {R})\), i.e.,
for any \(\phi \in W^{1,\infty }(\mathbb {R})\). To enhance clarity in distinguishing between the various variables in the functionals above, we have opted to explicitly write the variable x, even though it would be more accurate to omit x or replace x by “\(\cdot \)”. Moreover, since \(\partial _\xi y(t,\xi )>\ell \) for all \((t,\xi )\in [-T,T]\times \mathbb {R}\), we conclude that \(m(t,y(t,\cdot ))\), \(n(t,y(t,\cdot ))\in L^1(\mathbb {R})\) and \(\partial _t\partial _xu(t,y(t,\cdot ))\), \(\partial _t\partial _xv(t,y(t,\cdot ))\in L^1(\mathbb {R})\). Therefore, we have for any \(\phi \in W^{1,\infty }\)
and
Thus, the right-hand side of (3.52) and (3.53) exists, even though \(\partial _\xi y(t,\xi )\) is just a continuous function of \(\xi \). Taking into account that \(\phi ([y(t)]^{-1}(\cdot ))\in W^{1,\infty }(\mathbb {R})\) for all fixed \(t\in [-T,T]\), as soon as \(\phi \in W^{1,\infty }\), we can take \(\phi ([y(t)]^{-1}(x))\) instead of \(\phi (x)\) in (3.51). Changing the variables \(x=y(t,\xi )\), we obtain
which immediately implies (3.2) and thus (3.3).\(\square \)
Using Lemma 3.9, we can prove uniqueness of the solution of (1.6).
Proposition 3.10
(Uniqueness) The local solution (u, v) of the Cauchy problem (1.6) is unique in the class (3.36) for any \(k\in \mathbb {N}_0\).
Proof
Lemma 3.9 implies that such a solution (u, v) has the representation (3.4). Therefore, (3.10) for \(\zeta =y-\xi \) is equivalent to the Cauchy problem for (3.1) with initial data \(y(0,\xi )=y_0(\xi )\) (here y is the same as in Lemma 3.9). Since the vector field in (3.10) depends on the initial data \(m_0,n_0\) only and, according to Proposition 3.4, the Cauchy problem (3.10) has a unique solution, we conclude that the characteristic obtained in Lemma 3.9 is the same as that obtained in Proposition 3.4. This implies that any solution (u, v) in the considered class is that obtained in Proposition 3.8.\(\square \)
Proposition 3.11
Fix any two constants \(0<R_0< R\). Suppose that \(m_{0,j},n_{0,j}\in X^{k}\), \(j=1,2\), for some \(k\in \mathbb {N}_0\), are such that
Consider the two corresponding local solutions \((u_j,v_j)(t,x)\) of (1.6) in the class (3.36). Then the data-to-solution map satisfies the following conditions:
for some \(C_0=C_0(T,R_0)>0\), \(C=C(T,R)>0\) and sufficiently small \(T>0\).
Corollary 3.12
(Lipschitz continuity) It is easy to see that (3.54) imply the following Lipschitz property for the solutions \((m_j,n_j)(t,x)=(1-\partial _x^2)(u_j,v_j)(t,x)\), \(j=1,2\):
Proof
Consider the two characteristics \(\zeta _j(t,\xi )=y_j(t,\xi )-\xi \) obtained in Proposition 3.4, which correspond to \((m_{0,j}, n_{0,j})\), \(j=1,2\) (see (3.29) in Proposition 3.6). According to Propositions 3.8 and 3.10, the solutions \(u_j,v_j\) have the representation (3.4). Therefore, by introducing the integrals (cf. (3.37))
where
we have (cf. (3.38))
Using (cf. (3.39))
we conclude that (cf. (3.40))
First, let us show (3.54a) for \(C\left( [-T,T],C^1(\mathbb {R})\right) \). Observe that (we drop the arguments t, x of \(\breve{J}_{1,j}(t,x;m_{0,j})\) for simplicity)
Taking into account that (recall the definition of \(\xi _j\) given in (3.55))
we have for \(I_9\):
where we have used that \(\left| [y_j(t)]^{-1}(x)-x\right| \le \Vert \zeta _j(t,\cdot )\Vert _C\), \(j=1,2\). The integral \(I_{10}\) can be estimated as follows:
Here
and (recall (2.2b) and that \(|\xi _2-x|=\left| [y_2(t)]^{-1}(x)-x\right| \le \Vert \zeta _2(t,\cdot )\Vert _C\))
Combining (3.59), (3.60), (3.61), (3.26a) and (3.30a), we obtain
with \(T>0\) sufficiently small. Arguing similarly for \(\breve{J}_{2,1}(m_{0,1})-\breve{J}_{2,2}(m_{0,2})\) and \(\breve{J}_{i,1}(n_{0,1})-\breve{J}_{i,2}(n_{0,2})\), \(i=1,2\), we eventually arrive at
for all \((t,x)\in [-T,T]\times \mathbb {R}\) and for some \(C_0=C_0(T,R_0)>0\). Combining (3.56), (3.58) and (3.62), we obtain (3.54a) for \(C\left( [-T,T],C^1(\mathbb {R})\right) \).
To prove (3.54a) for \(C\left( [-T,T],W^{1,1}(\mathbb {R})\right) \), we observe that (cf. (3.59))
Changing the order of integration, we have the following estimate for \(I_{11}\):
As in (3.61), we can split the integral \(I_{12}\) as follows:
Here \(I_{12,1}\) and \(I_{12,2}\) can be estimated as follows:
and
Combining (3.63), (3.64), (3.65), (3.26a) and (3.30a), we obtain
where \(C_0=C_0(T,R_0)>0\). Arguing similarly for \(\breve{J}_{2,1}(m_{0,1})-\breve{J}_{2,2}(m_{0,2})\) and \(\breve{J}_{i,1}(n_{0,1})-\breve{J}_{i,2}(n_{0,2})\), \(i=1,2\), we eventually arrive at
for all \(t\in [-T,T]\) and some \(C_0=C_0(T,R_0)>0\). Combining (3.66) and (3.56), we obtain (3.54a) for \(C\left( [-T,T],W^{1,1}(\mathbb {R})\right) \) and thus we have proved (3.54a).
Then successively differentiating (3.58) with respect to x and using (3.57) together with (3.26), (3.30) and (3.62), (3.66), we eventually arrive at (3.54b).\(\square \)
Finally, combining Propositions 3.8, 3.10 and 3.11 we obtain the main result of this section about the local well-posedness of Cauchy problem (1.6) in the class \(C\left( [-T,T], X^{k+2}\right) \) with \(k\in \mathbb {N}_0\):
Theorem 3.13
(Local well-posedness) Suppose that \(u_0,v_0\in X^{k+2}\), \(k\in \mathbb {N}_0\) (see (2.1) for the definition of \(X^k\)). Then for a sufficiently small \(T>0\) there exists a unique solution (u, v)(t, x) of the Cauchy problem (1.6), which satisfies
Moreover, u and v can be found by (3.4a) and (3.4b), respectively, where the characteristics \(y(t,\xi )=\zeta (t,\xi )+\xi \) are given in Proposition 3.4.
Finally, the data-to-solution map is Lipschitz continuous. More precisely, for any constant \(R>0\) and two solutions \((u_j,v_j)\), \(j=1,2\), with initial data \((u_{0,j},v_{0,j})\) such that
where \((m_{0,j},n_{0,j})=(1-\partial _x^2)(u_{0,j},v_{0,j})\), we have
with \((m_{j},n_{j})=(1-\partial _x^2)(u_{j},v_{j})\) and some \(C=C(T,R)>0\). In addition, the solutions \((u_j,v_j)\) satisfy the continuity condition (3.54a).
4 Blow-up criteria
We can extend the local characteristics \(\zeta \) obtained in Proposition 3.4 to a maximal interval \(\bigl (-\tilde{T}_{\max },T_{\max }\bigr )\), where \(0<T_{\max },\tilde{T}_{\max }\le \infty \). This means that for any \(\tilde{T},T>0\) which satisfy \(-\tilde{T}_{\max }<-\tilde{T}<0\) and \(0<T<T_{\max }\), there exists \(\ell =\ell (T,\tilde{T})>0\) such that \(\zeta (t,\xi )\in C\bigl ([-\tilde{T},T],E_\ell \bigr )\) is a unique solution of the Cauchy problem (3.10). Of course, \(\zeta \) also satisfies the regularity and decay conditions (3.25) on the interval \([-\tilde{T},T]\) and thus this interval can be used in Proposition 3.8 for obtaining a unique solution (u, v) of (1.6) in \(C\bigl ([-\tilde{T},T], X^{k+2}\bigr )\). Moreover, in Theorem 4.5 we will prove that the maximal time of existence of the local solution (u, v) of the Cauchy problem for (1.6a) is precisely \(\bigl (-\tilde{T}_{\max },T_{\max }\bigr )\).
In the following proposition we give a criterion for the nonexistence of the global characteristics \(\zeta \) and establish its regularity and decay properties up to its maximal time of existence.
Proposition 4.1
(Characteristics on the maximal interval) Assume that \(m_0,n_0\) and \(y_0\) satisfy the same conditions as in Proposition 3.4. Consider \(\zeta (t,\xi )\) on the maximal interval \(\bigl (-\tilde{T}_{\max },T_{\max }\bigr )\), with \(0<T_{\max },\tilde{T}_{\max }\le \infty \). Then \(T_{\max }\) and/or \(\tilde{T}_{\max }\) are finite if and only if
Moreover, the characteristics \(\zeta (t,\xi )\) can be uniquely continued up to the blow-up time in such a way that it satisfies the following regularity and decay properties (cf. (3.25)):
for any closed and bounded \(\mathcal {I}\subset \overline{\bigl (-\tilde{T}_{\max }, T_{\max }\bigr )}\).
Finally, for all \(\xi ^\prime \) such that \(\partial _\xi y\left( T_{\max },\xi ^\prime \right) =0\) or \(\partial _\xi y\left( -\tilde{T}_{\max },\xi ^\prime \right) =0\) we have
Proof
The times \(T_{\max }\) and/or \(\tilde{T}_{\max }\) are finite if and only if either \(\Vert \zeta (t,\cdot )\Vert _{C^{1}(\mathbb {R})}\) blows up as \(t\rightarrow T_{\max }\) and/or \(t\rightarrow -\tilde{T}_{\max }\) or \(\inf \limits _{\xi \in \mathbb {R}}\partial _\xi y(t,\xi )= 1+\inf \limits _{\xi \in \mathbb {R}}\partial _\xi \zeta (t,\xi )\) converges to zero as \(t\rightarrow T_{\max }\) and/or \(t\rightarrow -\tilde{T}_{\max }\). Using (3.24), (3.27) and (3.8), we have the following a priori estimates for \(\zeta \) (cf. (3.26)):
for \(t\in \left( -\tilde{T}_{\max },T_{\max }\right) \). The latter estimates imply that \(\Vert \zeta (t,\cdot )\Vert _{C^{1}}\) cannot blow up in finite time, which implies the blow-up criteria (4.1).
The inequalities (4.4) also imply that the characteristics \(\zeta (t,\xi )\) and \(\partial _\xi \zeta (t,\xi )\) can be continued up to the finite \(T_{\max }\) and/or \(-\tilde{T}_{\max }\) by taking the limit in the variable t in (3.24) and (3.27) respectively. Then inequalities (3.8) as well as boundedness of \(\partial _\xi y_0,m_0,n_0\) on the line yield that \(\partial _t\zeta ,\zeta \in L^\infty (\mathcal {I},W^{1,\infty }(\mathbb {R}))\). Taking into account that \(\partial _\xi y_0,m_0,n_0\in C\) and that the functions under the integral in (3.27) are continuous and uniformly bounded with respect to \(\xi \) for all fixed \(t\in (-\tilde{T}_{\max },T_{\max })\) (see item 1 in Lemma 3.1 and (3.8)), we have by the Dominated Convergence Theorem that \(\zeta \in L^\infty (\mathcal {I},C^{1}(\mathbb {R}))\). Finally, in the case \(T_{\max }<\infty \) and \(T_{\max }\in \mathcal {I}\), we split up the integrals in (3.24) and (3.27) into a sum \(\int \limits _0^{T_{\max }-\varepsilon }\,\textrm{d}\eta +\int \limits _{T_{\max }-\varepsilon }^{T_{\max }}\,\textrm{d}\eta \) for some \(\varepsilon >0\) and conclude that \(\zeta \in C(\mathcal {I},C^{1}(\mathbb {R}))\). The case \(\tilde{T}_{\max }<\infty \) can be treated in a similar way.
Now we prove the decay properties of \(\zeta \) and \(\partial _t\zeta \). Arguing in the same way as in Lemma 3.3 with \(\mathcal {I}\) instead of \([-T,T]\) and \(\Vert \partial _t\zeta (t^*,\cdot )\Vert _{L^{\infty }}\) instead of \(\Vert \partial _t\zeta (t^*,\cdot )\Vert _C\) in (3.23), we conclude that \(J_j(m_0),J_j(n_0)\in C(\mathcal {I},L^1(\mathbb {R}))\) and \(J_j(|m_0|),J_j(|n_0|)\in C(\mathcal {I},L^1(\mathbb {R}))\). Taking into account that we have (3.15) for U, V and \(|W|\le \frac{1}{2}(J_1(|m_0|)+J_2(|m_0|))\), \(|Z|\le \frac{1}{2}(J_1(|n_0|)+J_2(|n_0|))\) for not strictly monotone increasing y, we conclude that \(U,W,V,Z\in L^\infty (\mathcal {I},L^1(\mathbb {R}))\). Therefore, (3.24) and (3.27) imply that \(\zeta \in C(\mathcal {I},W^{1,1}(\mathbb {R}))\) and \(\partial _t\zeta \in L^{\infty }(\mathcal {I},W^{1,1}(\mathbb {R}))\).
Finally, let us prove (4.3). Suppose that \(m_0(y_0(\xi ^\prime ))=n_0(y_0(\xi ^\prime ))=0\). Then (3.27) implies that \(\partial _\xi y(t,\xi ^\prime )=\partial _\xi y_0(\xi ^\prime )\) for all \(t\in (-\tilde{T}_{\max },T_{\max })\), which contradicts the assumption that \(\partial _\xi y(T_{\max },\xi ^\prime )=0\) or \(\partial _\xi y(-\tilde{T}_{\max },\xi ^\prime )=0\).\(\square \)
Remark 4.2
Notice that since \(\partial _\xi y(t,\xi )\rightarrow 1\), \(\xi \rightarrow \pm \infty \), for all \(t\in \mathcal {I}\), the functions \(\partial _\xi y(T_{\max },\xi )\) and/or \(\partial _\xi y(-\tilde{T}_{\max },\xi )\) can be zero at the finite \(\xi \) only.
Remark 4.3
Observe that the regularity and decay properties (4.2) of the characteristics on the time interval which can include the blow-up time, are weaker than that for the local characteristics, see (3.25). We lose the regularity because at \(t=T_{\max }\) and/or \(t=-\tilde{T}_{\max }\) the characteristics y are, in general, not strictly monotone increasing and thus \(W(t,\cdot ), Z(t,\cdot )\not \in C(\mathbb {R})\) and \(W,Z\not \in C(\mathcal {I}, L^\infty (\mathbb {R}))\) (cf. [12, Theorem 1.1, Item (i) and Lemma 3.1]).
Proof
Let us show that \(W(t,\cdot )\) and \(Z(t,\cdot )\) defined by (3.7) are, in general, discontinuous for not strictly monotone increasing \(y(t,\cdot )\). We give a proof for W, the function Z can be analyzed similarly. Suppose that \(y(t,\xi )\) is strictly monotone increasing for \(\xi \in (-\infty ,a)\cup (b,\infty )\) and it is constant for \(\xi \in [a,b]\). Denoting \(\tilde{m}_0(\xi )=m_0(y_0(\xi ))\partial _\eta y_0(\xi )\), we have from (3.7)
which implies that, in general, \(W(t,a-)\ne W(t,a+)\).
Now we prove that \(W\not \in C([t_0-\varepsilon ,t_0], L^\infty )\), for some \(t_0\in \mathbb {R}\), \(\varepsilon >0\), where \(y(t_0,\cdot )\) is strictly monotone increasing for \(\xi \in (-\infty ,a)\cup (b,\infty )\) and is constant for \(\xi \in [a,b]\), while \(y(t,\cdot )\) is strictly monotone increasing for all \(t\in [t_0-\varepsilon ,t_0)\) (the proof for Z is the same). For all \(\xi \in (a,b)\) we have
where (we drop the arguments of \(I_j\), \(j=13,14,15\) for simplicity)
Equations (4.5) imply that \(\Vert I_j(t,t_0,\cdot )\Vert _{L^\infty [a,b]}\rightarrow 0\) as \(t\rightarrow t_0\), \(j=13,14\), while \(\Vert I_{15}(t,t_0,\cdot )\Vert _{L^\infty [a,b]}\) has, in general, nonzero limit as \(t\rightarrow t_0\).\(\square \)
Remark 4.4
If the characteristics \(y(T_{\max },\cdot )\) and/or \(y(-\tilde{T}_{\max },\cdot )\) are strictly monotone increasing, then \(\zeta \) satisfies the regularity and decay properties (3.25) up to the blow-up time. The proof proceeds along the same lines as demonstrated in Proposition 3.4.
Now we can establish the blow-up criteria for the local solution of the Cauchy problem (1.6).
Theorem 4.5
(Blow-up criteria) Suppose that \(m_0,n_0\in X^0\). Consider \(\zeta (t,\xi )\), obtained in Proposition 4.1, on the maximal interval \(\bigl (-\tilde{T}_{\max },T_{\max }\bigr )\), with \(0<T_{\max },\tilde{T}_{\max }\le \infty \).
If \(T_{\max }\) and/or \(\tilde{T}_{\max }\) are finite, then we have
where \((m,n)(t,x)=(1-\partial _x^2)(u,v)(t,x)\) with (u, v)(t, x) being the unique solution of the Cauchy problem (1.6) in \(C\left( [-\tilde{T},T], X^0\right) \) for any \(-\tilde{T}_{\max }<-\tilde{T}<0\) and \(0<T<T_{\max }\).
Moreover, the following conditions are equivalent
-
(I)
\( \limsup \limits _{\begin{array}{c} t\rightarrow T_{\max },\,\,and/or\\ t\rightarrow -\tilde{T}_{\max } \end{array}} \left( \Vert m(t,\cdot )\Vert _{C}+\Vert n(t,\cdot )\Vert _{C} \right) =\infty ; \)
-
(II)
\( \lim \limits _{ \begin{array}{c} t\rightarrow T_{\max },\,\,and/or\\ t\rightarrow -\tilde{T}_{\max } \end{array}} \Bigl (\, \inf \limits _{\xi \in \mathbb {R}} (\partial _\xi y(t,\xi ))\Bigr )=0; \)
-
(III)
\(\limsup \limits _{ \begin{array}{c} t\rightarrow T_{\max },\,\,and/or\\ t\rightarrow -\tilde{T}_{\max } \end{array}} \Bigl (\, \sup \limits _{x\in \mathbb {R}}\left[ \left( (\partial _xu)n-un+(\partial _xv)m+vm \right) (t,x) \right] \Bigr )=\infty \);
-
(IV)
\(\int \limits _{0}^{T_{\max }} \Vert m(t,\cdot )\Vert _C+\Vert n(t,\cdot )\Vert _C\,\textrm{d}t=\infty \) and/or \(\int \limits _{0}^{-\tilde{T}_{\max }} \Vert m(t,\cdot )\Vert _C+\Vert n(t,\cdot )\Vert _C\,\textrm{d}t=\infty \).
Proof
Taking the characteristics y on the maximal interval \(\bigl (-\tilde{T}_{\max },T_{\max }\bigr )\) in the representation (3.4) (see Theorem 3.13), we obtain the local solution on the interval \([-\tilde{T},T]\) with any \(-\tilde{T}_{\max }<-\tilde{T}<0\) and \(0<T<T_{\max }\). Suppose that \(T_{\max }<\infty \). Remark 4.2 implies that there exists \(\xi ^\prime \in \mathbb {R}\) such that \(\partial _\xi y(T_{\max },\xi ^\prime )=0\). Since (u, v) admits the representation (3.4), the equalities (3.3) hold for all fixed \(t\in \bigl (-\tilde{T}_{\max }, T_{\max }\bigr )\) which, together with (4.3), imply that either \(|m(t,y(t,\xi ^\prime ))|\rightarrow \infty \) or \(|n(t,y(t,\xi ^\prime ))|\rightarrow \infty \) as \(t\rightarrow T_{\max }\). Arguing in the same way in the case \(\tilde{T}_{\max }<\infty \), we arrive at (4.6).
Now let us prove that the statements (I)–(IV) are equivalent. We will prove that (I) \(\Rightarrow \) (II) \(\Rightarrow \) (III) \(\Rightarrow \) (I) and (II) \(\Rightarrow \) (IV) \(\Rightarrow \) (I).
(I) \(\Rightarrow \) (II).
Since the right-hand side of (3.3) is finite for all \(t\in \bigl (-\tilde{T}_{\max }, T_{\max }\bigr )\), we conclude that (II) holds.
(II) \(\Rightarrow \) (III). Since \(\partial _ty(t,\xi )= \left( \partial _xu-u\right) (\partial _xv+v)(t,y(t,\xi ))\), we have
for \((t,\xi )\in \bigl (-\tilde{T}_{\max }, T_{\max }\bigr ) \times \mathbb {R}\). This implies that
which, together with (II), yields (III).
(III) \(\Rightarrow \) (I). This follows from
and (3.8).
(II) \(\Rightarrow \) (IV). Assume that \(T_{\max }<\infty \). Then from (4.7) we conclude that
Using (3.8) we obtain form (4.8)
Arguing similarly in the case \(\tilde{T}_{\max }<\infty \), we arrive at (IV).
(IV) \(\Rightarrow \) (I). Follows form the fact that \(\Vert m(t,\cdot )\Vert _C\) and \(\Vert n(t,\cdot )\Vert _C\) are finite for all \(t\in \bigl (-\tilde{T}_{\max }, T_{\max }\bigr )\).\(\square \)
Remark 4.6
The blow-up criteria established in Theorem 4.5 generalize [12, Theorem 3.2], where similar results were obtained for the Cauchy problem for the FORQ equation (where \(u=v\)) with initial data \(m_0\in X^k\), \(k\in \mathbb {N}\), having compact support. Also notice that Item (III) in Theorem 4.5 was previously obtained in [39, Theorem 4.2] for \(m(t,\cdot ),n(t,\cdot ) \in H^{s}(\mathbb {R})\), \(s>\frac{1}{2}\) (see also [35, Theorem 4.2] for the two-component system with high order nonlinearity and [15, Theorem 4.3] for the FORQ equation). Finally, for the solution \(m(t,\cdot ),n(t,\cdot )\in H^{s}(\mathbb {R})\), \(s>\frac{1}{2}\), it was established in [39, Theorem 4.1] (see also [35, Theorem 4.1] and [15, Theorem 4.2]), that if \(T_{\max }<\infty \), then
The latter condition is weaker than that in Theorem 4.5, Item (IV) obtained for \(m(t,\cdot ),n(t,\cdot )\in X^0\).
Remark 4.7
Theorem 4.5 implies that the maximal time interval \(\left( -\tilde{T}_{\max },T_{\max }\right) \) of the solution (u, v) with \(u_0,v_0\in X^{k+2}\), \(k\in \mathbb {N}_0\), does not depend on the regularity index k (cf. [39, Remark 4.1] and [15, Remark 4.1]). Indeed, consider the solution \((u^{\prime },v^{\prime })\) in \(X^{k^{\prime }+2}\), \(k^{\prime }\in \mathbb {N}_0\), \(k^{\prime }<k\), on the maximal interval \(\left( -\tilde{T}_{\max }^{\prime }, T_{\max }^{\prime }\right) \) with the same initial data \(u_0,v_0\in X^k\). Since \(k^{\prime }<k\) we have that \(\left( -\tilde{T}_{\max },T_{\max }\right) \subseteq \left( -\tilde{T}_{\max }^{\prime }, T_{\max }^{\prime }\right) \) and, due to the uniqueness, \((u,v)=(u^{\prime },v^{\prime })\) on \(\left( -\tilde{T}_{\max },T_{\max }\right) \). If, for example, \(T_{\max }<T_{\max }^{\prime }\), then \(u^{\prime },v^{\prime }\in C([0,T_{\max }],X^{k^{\prime }+2})\) and thus \(\Vert m(t,\cdot )\Vert _C+\Vert n(t,\cdot )\Vert _C<\infty \) as \(t\rightarrow T_{\max }\). Theorem 4.5 implies that \(\inf \limits _{\xi \in \mathbb {R}}\partial _\xi y(T_{\max },\xi )>0\) and therefore the solution u, v can be continued beyond \(T_{\max }\) in the class \(X^{k+2}\). Arguing similarly for \(\tilde{T}_{\max }^{\prime }\), we conclude that \(\tilde{T}_{\max }=\tilde{T}_{\max }^{\prime }\) and \(T_{\max }=T_{\max }^{\prime }\).
In conclusion of this section, we elucidate the local-in-space sufficient condition that precipitates the finite-time blow-up of the solution pair (u, v). This condition was initially identified in the context of the two-component system (1.1), accommodating initial data \(m_0,n_0\) in \(H^s\cap L^1\) for \(s>\frac{1}{2}\), as demonstrated in [39, Theorem 4.3]. Subsequent corroborations and extensions of this result can be found in [11, Theorem 5.1], [12, Theorem 4.2], [35, Theorem 5.1], and [15, Theorem 5.2]. We extend these findings to solutions in the space \(X^{k}\), see (2.1).
Theorem 4.8
[39, Theorem 4.3]. Assume that \(m_0,n_0\in X^k\), \(k\in \mathbb {N}_0\), \(m_0(x),n_0(x)\ge 0\) for all \(x\in \mathbb {R}\) and there exists \(x_0\in \mathbb {R}\) such that \(m_0(x_0),n_0(x_0)>0\). Consider the corresponding solution (u, v)(t, x) of (1.6) on the maximal interval \(\left( -\tilde{T}_{\max },T_{\max }\right) \) and let
where
Then we have
-
if \(M_0<-\sqrt{2L_0N_0}\), the maximal existence time \(T_{\max }>0\) is finite, and it has the following upper bound:
$$\begin{aligned} T_{\max }\le t_1. \end{aligned}$$In the case \(T_{\max }=t_1\), we have the following estimates for the blow-up rate:
$$\begin{aligned} \Vert m(t,\cdot )\Vert _{C}+\Vert n(t,\cdot )\Vert _C\ge \frac{2}{t_2L_0(T_{\max }-t)},\quad t\in \left( 0,T_{\max }\right) , \end{aligned}$$(4.10)and
$$\begin{aligned} \inf \limits _{\xi \in \mathbb {R}}\partial _\xi y(t,\xi ) \le t_2\frac{L_0}{2}(m_0+n_0)(x_0)(T_{\max }-t), \quad t\in \left( 0,T_{\max }\right) . \end{aligned}$$(4.11) -
If \(M_0>\sqrt{2L_0N_0}\), the maximal existence time \(-\tilde{T}_{\max }<0\) is finite and it has the following lower bound:
$$\begin{aligned} -\tilde{T}_{\max }\ge t_2. \end{aligned}$$In the case \(\tilde{T}_{\max }=t_2\), we have the following estimates for the blow-up rate:
$$\begin{aligned} \Vert m(t,\cdot )\Vert _{C}+\Vert n(t,\cdot )\Vert _C\ge \frac{2}{|t_1|L_0(t+\tilde{T}_{\max })},\quad t\in \bigl (-\tilde{T}_{\max },0\bigr ), \end{aligned}$$and
$$\begin{aligned} \inf \limits _{\xi \in \mathbb {R}}\partial _\xi y(t,\xi ) \le |t_1|\frac{L_0}{2}(m_0+n_0)(x_0) (t+\tilde{T}_{\max }), \quad t\in \bigl (-\tilde{T}_{\max },0\bigr ). \end{aligned}$$
Proof
The proof closely follows the methodology in [39], with minor modifications tailored to our specific context. Here, we provide a concise overview of the essential steps, highlighting where our approach diverges from that of [39]. Taking into account Remark 4.7, we can assume that \(k\ge 3\). Let us take \(y_0(\xi )=\xi \) and denote
Direct calculations show that, cf. [39, Lemma 4.5] (here we drop the arguments of M, u and v for simplicity)
Then arguing similarly as in [39, Theorem 4.3, equation (4.38)], we conclude that
and (see [39, equation (4.39)])
From the assumptions of the theorem and (3.3), \(N(t,y(t,x_0))>0\) for all \(t\in \bigl (-\tilde{T}_{\max },T_{\max }\bigr )\). Combining (4.12) and (4.13), we conclude that
and thus
Integrating the latter from 0 to t with \(t>0\), we obtain
Combining (4.13) and (4.15), we obtain
which, after integration from 0 to t, \(t>0\), leads to
where \(t_1\) and \(t_2\) are the solutions of the quadratic equation \(t^2+\frac{2M_0}{L_0N_0}t+\frac{2}{L_0N_0}=0\) given in (4.9). In view of the assumption \(M_0<-\sqrt{2L_0N_0}\), we have that \(0<t_1<t_2\) which, together with (4.17), implies that \(T_{\max }\le t_1\) and \(\Vert m(t,\cdot )\Vert _C+\Vert n(t,\cdot )\Vert _C\rightarrow \infty \) as \(t\rightarrow T_{\max }\). The blow-up rate (4.10) follows from (4.17) and the inequality
while the estimate (4.11) follows from (4.17) and (see (3.3); cf. [12, Theorem 4.2])
Arguing similarly for \(-\tilde{T}_{\max }\), where we integrate (4.14) and (4.16) from t to 0 with \(t<0\), we obtain the lower bound for \(-\tilde{T}_{\max }\) as well as the blow-up rate.\(\square \)
References
Ablowitz, M.J., Musslimani, Z.H.: Integrable nonlocal nonlinear Schrödinger equation. Phys. Rev. Lett. 110, 064105 (2013)
Ablowitz, M.J., Musslimani, Z.H.: Integrable nonlocal nonlinear equations. Stud. Appl. Math. 139, 7–59 (2017)
Anco, S.C., Recio, E.: A general family of multi-peakon equations and their properties. J. Phys. A Math. Theor. 52, 125203 (2019)
Boutet de Monvel, A., Karpenko, I., Shepelsky, D.: A Riemann–Hilbert approach to the modified Camassa-Holm equation with nonzero boundary conditions. J. Math. Phys. 61, 031504 (2020)
Bressan, A., Constantin, A.: Global conservative solutions of the Camassa–Holm equation. Arch. Ration. Mech. Anal. 183, 215–239 (2007)
Bressan, A., Constantin, A.: Global dissipative solutions of the Camassa–Holm equation. Anal. Appl. (Singap.) 5, 1–27 (2007)
Chang, X.-K., Hu, X.-B., Szmigielski, J.: Multipeakons of a two-component modified Camassa–Holm equation and the relation with the finite Kac–van Moerbeke lattice. Adv. Math. 299, 1–35 (2016)
Chen, R.M., Guo, F., Liu, Y., Qu, C.: Analysis on the blow-up of solutions to a class of integrable peakon equations. J. Funct. Anal. 270(6), 2343–2374 (2016)
Fokas, A.S.: On a class of physically important integrable equations. Physica D 87, 145–150 (1995)
Fuchssteiner, B.: The Lie algebra structure of nonlinear evolution equations admitting infinite dimensional Abelian symmetry groups. Prog. Theoret. Phys. 65, 861–876 (1981)
Fu, Y., Gui, G., Liu, Y., Qu, C.: On the Cauchy problem for the integrable modified Camassa–Holm equation with cubic nonlinearity. J. Differ. Equ. 255, 1905–1938 (2013)
Gao, Y., Liu, J.-G.: The modified Camassa–Holm equation in Lagrangian coordinates. Discrete Contin. Dyn. Syst. Ser. B. 23(6), 2545–2592 (2018)
Gao, Y., Liu, J.-G.: Global convergence of a sticky particle method for the modified Camassa–Holm equation. SIAM J. Math. Anal. 49, 1267–1294 (2017)
Grunert, K., Holden, H., Raynaud, X.: Global conservative solutions to the Camassa–Holm equation for initial data with nonvanishing asymptotics. Discrete Contin. Dyn. Syst. 32(12), 4209–4227 (2012)
Gui, G., Liu, Y., Olver, P.J., Qu, C.: Wave-breaking and peakons for a modified Camassa–Holm equation. Commun. Math. Phys. 319, 731–759 (2013)
Himonas, A., Mantzavinos, D.: The Cauchy problem for the Fokas-Olver-Rosenau-Qiao equation. J. Nonlinear Anal. Theory Methods Appl. 95, 499–529 (2014)
Himonas, A., Mantzavinos, D.: Hölder continuity for the Fokas-Olver-Rosenau-Qiao equation. J. Nonlinear. Sci. 24, 1105–1124 (2014)
Holden, H., Raynaud, X.: Global conservative solutions of the Camassa–Holm equation—a Lagrangian point of view. Commun. Partial Differ. Equ. 32, 1511–1549 (2007)
Holden, H., Raynaud, X.: Dissipative solutions for the Camassa–Holm equation. Discrete Contin. Dyn. Syst. 24, 1047–1112 (2009)
Hou, Y., Fan, E., Qiao, Z.: The algebro-geometric solutions for the Fokas-Olver-Rosenau-Qiao (FORQ) hierarchy. J. Geom. Phys. 117, 105–133 (2017)
Karlsen, K.H., Rybalko, Ya.: On the well-posedness of a nonlocal (two-place) FORQ equation via a two-component peakon system. J. Math. Anal. Appl. 529, 127601 (2024)
Karpenko, I.: Long-time asymptotics for the modified Camassa–Holm equation with nonzero boundary conditions. J. Math. Phys. Anal. Geom. 16(4), 418–453 (2022)
Lou, S.Y., Huang, F.: Alice–Bob physics: coherent solutions of nonlocal KdV systems. Sci. Rep. 7, 869 (2017)
Lou, S.Y., Qiao, Z.: Alice–Bob peakon systems. Chin. Phys. Lett. 34(10), 100201 (2017)
Matsuno, Y.: Smooth and singular multisoliton solutions of a modified Camassa–Holm equation with cubic nonlinearity and linear dispersion. J. Phys. A Math. Theor. 47, 125203 (2014)
Mi, Y., Mu, C.: Well-posedness and analyticity for an integrable two-component system with cubic nonlinearity. J. Hyperbolic Differ. Equ. 10(04), 703–723 (2013)
Novikov, V.: Generalizations of the Camassa–Holm equation. J. Phys. A Math. Theor. 42, 342002 (2009)
Olver, P.J., Rosenau, P.: Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support. Phys. Rev. E 53, 1900–1906 (1996)
Qiao, Z.J.: A new integrable equation with cuspons and W/M-shape-peaks solitons. J. Math. Phys. 47, 112701 (2006)
Rybalko, Ya., Shepelsky, D.: Long-time asymptotics for the integrable nonlocal focusing nonlinear Schrödinger equation for a family of step-like initial data. Commun. Math. Phys. 382(1), 87–121 (2021)
Schiff, J.: Zero curvature formulations of dual hierarchies. J. Math. Phys. 37, 1928 (1996)
Schäfer, T., Wayne, C.E.: Propagation of ultra-short optical pulses in cubic nonlinear media. Physica D 196, 90–105 (2004)
Song, J.F., Qu, C.Z., Qiao, Z.J.: A new integrable two-component system with cubic nonlinearity. J. Math. Phys. 52, 013503 (2011)
Tian, K., Liu, Q.P.: Tri-Hamiltonian duality between the Wadati-Konno-Ichikawa hierarchy and the Song-Qu-Qiao hierarchy. J. Math. Phys. 54, 043513 (2013)
Wang, Z., Yan, K.: Blow-up data for a two-component Camassa–Holm system with high order nonlinearity. J. Differ. Equ. 358(15), 256–294 (2023)
Xia, B., Qiao, Z., Zhou, R.: A synthetical two-component model with peakon solutions. Stud. Appl. Math. 135(3), 248–276 (2015)
Wang, Y., Zhu, M.: On the Cauchy problem for a two-component peakon system with cubic nonlinearity. J. Dyn. Differ. Equ. (2022). https://doi.org/10.1007/s10884-022-10187-6
Yan, K.: On the blow up solutions to a two-component cubic Camassa–Holm system with peakons. Discrete Contin. Dyn. Syst. 40(7), 4565–4576 (2020)
Yan, K., Qiao, Z., Zhang, Y.: Blow-up phenomena for an integrable two-component Camassa–Holm system with cubic nonlinearity and peakon solutions. J. Differ. Equ. 259(11), 6644–6671 (2015)
Yang, S.: Blow-up phenomena for the generalized FORQ/MCH equation. Z. Angew. Math. Phys. (2020). https://doi.org/10.1007/s00033-019-1241-9
Yang, S., Chen, J.: On the finite time blow-up for the high-order Camassa-Holm-Fokas-Olver-Rosenau-Qiao equations. J. Differ. Equ. 379, 829–861 (2024)
Zeng, F., Gao, Y., Xue, X.: Global weak solutions to the generalized mCH equation via characteristics. Discrete Contin. Dyn. Syst. Ser. B 27(8), 4317–4329 (2022)
Zhang, Q.: Global wellposedness of cubic Camassa–Holm equations. Nonlinear Anal. 133, 61–73 (2016)
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Karlsen, K.H., Rybalko, Y. On the well-posedness of the Cauchy problem for the two-component peakon system in \(C^k\cap W^{k,1}\). Z. Angew. Math. Phys. 75, 113 (2024). https://doi.org/10.1007/s00033-024-02246-3
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DOI: https://doi.org/10.1007/s00033-024-02246-3
Keywords
- FORQ equation
- Two-component peakon equation
- Nonlocal (Alice–Bob) integrable system
- Cubic nonlinearity
- Local well-posedness
- Blow-up criteria