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1.
We first prove the existence of global conservative solutions to the Cauchy problem for a modified two-component Camassa-Holm shallow water system. Then, we show that these global solutions, which depend continuously on the initial date, construct a semigroup.  相似文献   

2.
We prove the existence of global weak solutions for a new periodic integrable equation with peakon solutions.  相似文献   

3.
4.
This paper is concerned with global existence and blow-up phenomena for an integrable two-component Camassa-Holm shallow water system. A new global existence result and several new blow-up results of strong solutions to the system are presented. Our obtained results for the system are sharp and improve considerably earlier results.  相似文献   

5.
This paper is concerned with global existence and blow-up phenomena for the weakly dissipative Camassa-Holm equation. A new global existence result and a new blow-up result for strong solutions to the equation with certain profiles are presented. The obtained results improve considerable the previous results.  相似文献   

6.
A nonlinear shallow water equation, which includes the famous Camassa-Holm (CH) and Degasperis-Procesi (DP) equations as special cases, is investigated. The local well-posedness of solutions for the nonlinear equation in the Sobolev space Hs(R) with is developed. Provided that does not change sign, u0Hs () and u0L1(R), the existence and uniqueness of the global solutions to the equation are shown to be true in u(t,x)∈C([0,∞);Hs(R))∩C1([0,∞);Hs−1(R)). Conditions that lead to the development of singularities in finite time for the solutions are also acquired.  相似文献   

7.
We establish the local well-posedness for a new nonlinearly dispersive wave equation which has solutions that exist for indefinite times as well as solutions that blowup infinite time. We also derive an explosion criterion for the equation, and we give a sharp estimate of the existence time for solutions with smooth initial data.  相似文献   

8.
We consider higher-order Camassa-Holm equations describing exponential curves of the manifold of smooth orientation-preserving diffeomorphisms of the unit circle in the plane. We establish the existence of global weak solutions. We also present some invariant spaces under the action of the equation. Moreover, we prove a “weak equals strong” uniqueness result.  相似文献   

9.
10.
We prove uniqueness within a class of discontinuous solutions to the nonlinear and third order dispersive Degasperis-Procesi equation
  相似文献   

11.
We establish the local well-posedness for a generalized Dullin-Gottwald-Holm equation by using Kato’s theory. Furthermore, the orbital stability of the peaked solitary waves is also proved.  相似文献   

12.
We prove a blow-up result for a nonlinear shallow water equation by showing that certain initial profiles evolve into breaking waves.  相似文献   

13.
We consider global solutions of a dynamical equation in ferrimagnet. We show that it admits a global weak solution by using the penalty method. By the energy estimates method we show there exists a unique global smooth solution. Finally we establish the relationship between this equation and wave maps.  相似文献   

14.
In the paper, we first show the existence of global periodic conservative solutions to the Cauchy problem for a periodic modified two-component Camassa-Holm equation. Then we prove that these solutions, which depend continuously on the initial data, construct a semigroup.  相似文献   

15.
We present a blow-up criterion for the periodic Camassa-Holm equation. The condition obtained for blow-up uses two of the conservation laws associated with the equation and improves upon some recent results.Received: 18 June 2004  相似文献   

16.
We establish the local well-posedness for the generalized Camassa–Holm equation. We also prove that the equation has smooth solutions that blow up in finite time.  相似文献   

17.
This paper deals with the Cauchy problem for a higher order shallow water equation yt+auxy+buyx=0, where and k=2. The local well-posedness of solutions for the Cauchy problem in Sobolev space Hs(R) with s?7/2 is obtained. Under some assumptions, the existence and uniqueness of the global solutions to the equation are shown, and conditions that lead to the development of singularities in finite time for the solutions are also acquired. Finally, the weak solution for the equation is considered.  相似文献   

18.
In this paper, we prove the existence of global weak solution for an integrable two-component Camassa-Holm shallow water system provided the initial data satisfying some certain conditions.  相似文献   

19.
A generalization of the Camassa-Holm equation, a model for shallow water waves, is investigated. Using the pseudoparabolic regularization technique, its local well-posedness in Sobolev space Hs(R) with is established via a limiting procedure. In addition, a sufficient condition for the existence of weak solutions of the equation in lower order Sobolev space Hs with is developed.  相似文献   

20.
We prove endpoint Strichartz estimates for the Klein-Gordon and wave equations in mixed norms on the polar coordinates in three spatial dimensions. As an application, global wellposedness of the nonlinear Dirac equation is shown for small data in the energy class with some regularity assumption for the angular variable.  相似文献   

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