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In this paper, we are concerned with the finite time blow-up for the high-order Camassa-Holm-Fokas-Olver-Rosenau-Qiao equations, which is a generalization of the Camassa-Holm equation and the Fokas-Olver-Rosenau-Qiao equation. We explore how high-order nonlinearities affect the dispersive dynamics and breakdown mechanism of solutions. Firstly, we established the local well-posedness for the Cauchy problem in the framework of Besov spaces. Then, we derive the precise blow-up mechanism for strong solutions by means of the transport equation theory. Finally, a sufficient condition on initial data that leads to the finite time blow-up of the second-order derivative of the solutions is described in detail. [ABSTRACT FROM AUTHOR]

For the famous Camassa-Holm equation, the well-posedness in C ([ 0 , T ] ; B p , 1 1 + 1 p (R)) with 1 ≤ p ≤ 2 and the ill-posedness in C ([ 0 , T ] ; B ∞ , 1 1 (R)) had been studied in [15,16,23]. That is to say, it left an open problem in the critical case C ([ 0 , T ] ; B p , 1 1 + 1 p (R)) with 2 < p < + ∞ proposed by Danchin in [15,16]. In this paper, we solve this problem and obtain the local well-posedness for the Camassa-Holm equation in critical Besov spaces C ([ 0 , T ] ; B p , 1 1 + 1 p (R)) with 1 ≤ p < + ∞. It is worth mentioning that our method is suitable for many Camassa-Holm type equations, such as the Novikov equation and the two-component Camassa-Holm system, and can also improve their index of the local well-posedness. [ABSTRACT FROM AUTHOR]

In this paper, we mainly investigate the Cauchy problem of the cubic Camassa-Holm-type equations which arise as an asymptotic model with a nonlocal cubic nonlinearity for the unidirectional propagation of shallow water waves, which admit the single peaked solitons and multi-peakon solutions. We first establish the local well-posedness for the Cauchy problem of the new shallow-water model. Then, by overcoming the difficulties cased by the complicated mixed nonlinear structure, we present a precise blow-up scenario. Moreover, based on a generalized Ovsyannikov type theorem, we prove the existence and uniqueness of solutions for Cauchy problem in the Gevrey-Sobolev spaces with the lower bound of the lifespan, and show the continuity of the data-to-solution map. Finally, we investigate the persistence properties of the solution to the Cauchy problem, we prove that the solution maintains the corresponding properties at infinity within its lifespan provided the initial data decay exponentially and algebraically, respectively. [ABSTRACT FROM AUTHOR]

In this paper we mainly investigate the Cauchy problem of the Doi-Edwards polymer model with dimension d ≥ 2. The model was derived in the late 1970s to describe the dynamics of polymers in melts. The system contains a Navier-Stokes equation with an additional stress tensor which depend on the deformation gradient tensor and the memory function. The deformation gradient tensor satisfies a transport equation and the memory function satisfies a degenerate parabolic equation. We first proved the local well-posedness for the Doi-Edwards polymer model in Besov spaces by using the Littlewood-Paley theory. Moreover, if the initial velocity and the initial memory is small enough, we obtain a global existence result. [ABSTRACT FROM AUTHOR]

In this paper, we consider the well-posedness of the inhomogeneous nonlinear biharmonic Schrödinger equation with spatial inhomogeneity coefficient K (x) behaves like | x | − b for 0 < b < min ⁡ { N 2 , 4 }. We show the local well-posedness in the whole H s -subcritical case, with 0 < s ≤ 2. The difficulties of this problem come from the singularity of K (x) and the lack of differentiability of the nonlinear term. To resolve this, we derive the bilinear Strichartz's type estimates for the nonlinear biharmonic Schrödinger equations in Besov spaces. [ABSTRACT FROM AUTHOR]

Considered in this paper is the new shallow-water model with cubic nonlinearity, which admits the single peaked solitons and multi-peakon solutions, and includes both the modified Camassa-Holm equation (also called Fokas-Olver-Rosenau-Qiao equation) and the Novikov equation as two special cases. The main investigation is the Cauchy problem of the new shallow-water model with qualitative properties of its solutions. We first establish the local well-posedness for the Cauchy problem of the new shallow-water model, and then derive a precise blow-up scenario and the blow-up mechanisms for solutions with certain initial profiles. Moreover, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time. Finally, we investigate the persistence properties of the solution to the Cauchy problem of the new shallow-water model. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonlinear evolution equation modelling the propagation of surface waves in the shallow water regime of large amplitude, which is characterised by some cubical nonlinearities. First, we establish the local well-posedness in Besov space B 2 , 1 3 / 2. Then, we give a blow-up criterion. Finally, with a given analytic initial data, we establish the analyticity of the solutions in both variables, globally in space and locally in time. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we are concerned with the Cauchy problem of the generalized Camassa–Holm equation. Using a Galerkin-type approximation scheme, it is shown that this equation is well-posed in Sobolev spaces H s , s > 3 / 2 for both the periodic and the nonperiodic case in the sense of Hadamard. That is, the data-to-solution map is continuous. Furthermore, it is proved that this dependence is sharp by showing that the solution map is not uniformly continuous. The nonuniform dependence is proved using the method of approximate solutions and well-posedness estimates. Moreover, it is shown that the solution map for the generalized Camassa–Holm equation is Hölder continuous in H r -topology. Finally, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time. [ABSTRACT FROM AUTHOR]

In this paper we mainly investigate the Cauchy problem of a three-component Camassa–Holm system. We first prove the local well-posedness of the system in Besov spaces B p , r s with p , r ∈ [ 1 , ∞ ] , s > max ⁡ { 1 p , 1 2 } by using the Littlewood–Paley theory and transport equations theory. Then, we establish two blow-up criteria which along with the conservation laws enable us to study global existence. Moreover, if the initial data satisfies some certain sign conditions, we obtain a global existence result. Finally, we verify that the system possesses peakon solutions. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with the Cauchy problem of a model equation for shallow water waves of moderate amplitude, which was proposed by A. Constantin and D. Lannes [The hydrodynamical relevance of the Camassa–Holmand Degasperis–Procesi equations, Arch. Ration. Mech. Anal. 192 (2009) 165–186]. First, the local well-posedness of the model equation is obtained in Besov spaces , , (which generalize the Sobolev spaces ) by using Littlewood–Paley decomposition and transport equation theory. Second, the local well-posedness in critical case (with , , ) is considered. Moreover, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time. Finally, persistence properties on strong solutions are also investigated. [Copyright &y& Elsevier]

Abstract: Considered in this paper is the modified Camassa–Holm equation with cubic nonlinearity, which is integrable and admits the single peaked solitons and multi-peakon solutions. The short-wave limit of this equation is known as the short-pulse equation. The main investigation is the Cauchy problem of the modified Camassa–Holm equation with qualitative properties of its solutions. It is firstly shown that the equation is locally well-posed in a range of the Besov spaces. The blow-up scenario and the lower bound of the maximal time of existence are then determined. A blow-up mechanism for solutions with certain initial profiles is described in detail and nonexistence of the smooth traveling wave solutions is also demonstrated. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the Cauchy problem for a two-component Degasperis–Procesi system. Firstly, the local well-posedness for this system in the nonhomogeneous Besov spaces is established. Then the precise blow-up scenario for strong solutions to the system is derived. Finally, two new blow-up criterions and the exact blow-up rate of strong solutions to the system are presented. [Copyright &y& Elsevier]