On the Cauchy problem for a generalized Degasperis-Procesi equation

ABSTRACT

We first establish the local well-posedness for a generalized Degasperis-Procesi equation in nonhomogeneous Besov spaces. Then we present a global existence result for the equation. Moreover, we obtain a blow-up criteria and provide a sufficient condition for strong solutions to blow up in finite time.     

On the Cauchy problem and peakons of a two-component Novikov system

We study a two-component Novikov system, which is integrable and can be viewed as a twocomponent generalization of the Novikov equation with cubic nonlinearity. The primary goal of this paper is to understand how multi-component equations, nonlinear dispersive terms and other nonlinear terms affect the dispersive dynamics and the structure of the peaked solitons. We establish the local well-posedness of the Cauchy problem in Besov spaces B_(p,r)~s with 1 p, r +∞, s max{1 + 1/p, 3/2} and Sobolev spaces Hs(R)with s 3/2, and the method is based on the estimates for transport equations and new invariant properties of the system. Furthermore, the blow-up and wave-breaking phenomena of solutions to the Cauchy problem are studied. A blow-up criterion on solutions of the Cauchy problem is demonstrated. In addition, we show that this system admits single-peaked solitons and multi-peaked solitons on the whole line, and the single-peaked solitons on the circle, which are the weak solutions in both senses of the usual weak form and the weak Lax-pair form of the system.     

The Cauchy problem for the Novikov equation

In this paper we consider the Cauchy problem for the Novikov equation. We prove that the Cauchy problem for the Novikov equation is not locally well-posed in the Sobolev spaces ${H^s(\mathfrak{R})}$ with ${s < \frac{3}{2}}$ in the sense that its solutions do not depend uniformly continuously on the initial data. Since the Cauchy problem for the Novikov equation is locally well-posed in ${H^{s}(\mathfrak{R})}$ with s > 3/2 in the sense of Hadamard, our result implies that s =  3/2 is the critical Sobolev index for well-posedness. We also present two blow-up results of strong solution to the Cauchy problem for the Novikov equation in ${H^{s}(\mathfrak{R})}$ with s > 3/2.     

On the Cauchy problem of a two-component b-family system

In this paper, we study the Cauchy problem of a two-component b-family system. We first establish the local well-posedness for a two-component b-family system by Kato’s semigroup theory. Then, we derive the precise blow-up scenario for strong solutions to the system. Moreover, we present several blow-up results for strong solutions to the system.     

Analyticity of the Cauchy problem for two-component Hunter-Saxton systems

This paper is mainly concerned with the periodic Cauchy problem for a generalized two-component μ-Hunter-Saxton system with analytic initial data. The analyticity of its solutions is proved in both variables, globally in space and locally in time. The obtained result can be also applied to its special cases—the classical integrable two-component Hunter-Saxton system, the generalized μ-Hunter-Saxton equation and the classical Hunter-Saxton equation.     

Well-posedness and analyticity of the Cauchy problem for the multi-component Novikov equation

In this paper, we are concerned with the Cauchy problem of the multi-component Novikov equation. We establish the local well-posedness in a range of the Besov spaces by using Littlewood–Paley decomposition and transport equation theory. Moreover, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time.     

A note on ill-posedness of the Cauchy problem for Heisenberg wave maps

We introduce a notion of wave maps with a target in the sub-Riemannian Heisenberg group and study their relation with Riemannian wave maps with range in Lagrangian submanifolds. As an application we establish existence and eventually ill-posedness of the corresponding Cauchy problem.

     

A note on a modified two-component Camassa-Holm system

We consider the asymptotic behavior of solutions for a modified two-component Camassa-Holm (MCH2) system which arises in shallow water theory. It is proved that the corresponding solution to initial data with algebraic decay at infinity will retain this property in its lifespan.     

On the Cauchy problem for the periodic generalized Degasperis-Procesi equation

We mainly study the Cauchy problem of the periodic generalized Degasperis-Procesi equation. First, we establish the local well-posedness for the equation. Second, we give the precise blow-up scenario, a conservation law and prove that the equation has smooth solutions which blow up in finite time. Finally, we investigate the blow-up rate for the blow-up solutions.     

The global weak solutions to the Cauchy problem of the generalized Novikov equation

In this paper, we study the Cauchy problem of the generalized Novikov equation. We first show that under suitable condition, the strong solution exists globally via some a priori estimates. Then, we prove the existence and uniqueness of global weak solutions by the approximation method. We also obtain the exact peaked solutions.     

The Cauchy problem for an integrable two-component model with peakon solutions

In this paper, we are concerned with the Cauchy problem of the new integrable two-componnt system with cubic non-linearity, which was proposed by Xia, Qiao and Zhou. We establish the local well-posedness in a range of the Besov spaces. Moreover, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time.     

A note on a Cauchy problem for the Laplace equation: Regularization and error estimates

In this paper, a Cauchy problem for the Laplace equation is investigated. Based on the fundamental solution to the elliptic equation, we propose to solve this problem by the truncation method, which generates well-posed problem. Then the well posedness of the proposed regularizing problem and convergence property of the regularizing solution to the exact one are proved. Error estimates for this method are provided together with a selection rule for the regularization parameter. The numerical results show that our proposed numerical methods work effectively. This work extends to earlier results in Qian et al. (2008) [14] and Hao et al. (2009) [5].