一类弱耗散双组份Hunter-Saxton系统的爆破与爆破率

研究了一类周期弱耗散双组份Hunnter-Saxton系统的爆破现象.首先,给出了此类Hunnter-Saxton系统解的局部适定性及其精确的爆破机制;其次,证明了在一定的初始值下Hunnter-Saxton系统强解的几个爆破结果;最后,给出了HunnterSaxton系统强解的爆破率.     

BLOW-UP SOLUTIONS FOR A CASE OF b-FAMILY EQUATIONS

In this article, we study the blow-up solutions for a case of b-family equations.Using the qualitative theory of differential equations and the bifurcation method of dynamical systems, we obtain five types of blow-up solutions: the hyperbolic blow-up solution, the fractional blow-up solution, the trigonometric blow-up solution, the first elliptic blow-up solution, and the second elliptic blow-up solution. Not only are the expressions of these blow-up solutions given, but also their relationships are discovered. In particular, it is found that two bounded solitary solutions are bifurcated from an elliptic blow-up solution.     

INVARIANT SUBSPACES AND GENERALIZED FUNCTIONAL SEPARABLE SOLUTIONS TO THE TWO-COMPONENT b-FAMILY SYSTEM

Invariant subspace method is exploited to obtain exact solutions of the twocomponent b-family system.It is shown that the two-component b-family system admits the generalized functional separable solutions.Furthermore,blow up and behavior of those exact solutions are also investigated.     

On the Cauchy problem for a two-component Degasperis–Procesi system

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.     

Well-posedness,blow-up phenomena and persistence properties for a two-component water wave system

We first establish the local well-posedness for the Cauchy problem of a two-component water waves system in nonhomogeneous Besov spaces using the Littlewood–Paley theory. Then, we derive three new blow-up results for strong solutions to the system. Finally, we present two persistence properties for strong solutions to the system.     

Wave breaking and global existence for the generalized periodic two-component Hunter–Saxton system

In this paper, we study the wave-breaking phenomena and global existence for the generalized two-component Hunter–Saxton system in the periodic setting. We first establish local well-posedness for the generalized two-component Hunter–Saxton system. We obtain a wave-breaking criterion for solutions and results of wave-breaking solutions with certain initial profiles. We also determine the exact blow-up rate of strong solutions. Finally, we give a sufficient condition for global solutions.     

On the Cauchy problem for the two-component Camassa�CHolm system

In this paper we establish the local well-posedness for the two-component Camassa?CHolm system in a range of the Besov spaces. We also derive a wave-breaking mechanism for strong solutions. In addition, we determine the exact blow-up rate of such solutions to the system.     

Well-posedness and analytic solutions of the two-component Euler–Poincaré system

We first establish the local well-posedness for the Cauchy problem of the two-component Euler–Poincaré system in nonhomogeneous Besov spaces. Then, we derive a blow-up criterion for strong solutions to the system. Finally, we prove the existence of analytic solutions to the system.     

Well-posedness and blow-up solution for a modified two-component periodic Camassa–Holm system with peakons

Considered herein is a modified two-component periodic Camassa–Holm system with peakons. The local well-posedness and low regularity result of solutions are established. The precise blow-up scenarios of strong solutions and several results of blow-up solutions with certain initial profiles are described in detail and the exact blow-up rate is also obtained.     

Global existence and blow-up phenomena for an integrable two-component Camassa-Holm shallow water system

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.     

Blow-up for a weakly dissipative modified two-component Camassa–Holm system

In this paper, we study the Cauchy problem of a weakly dissipative modified two-component Camassa–Holm (MCH2) system. We first derive the precise blow-up scenario and then give several criteria guaranteeing the blow-up of the solutions. We finally discuss the blow-up rate of the blowing-up solutions.     

Blow-up phenomena and peakons for the b-family of FORQ/MCH equations

This paper is devoted to studying the modified b-family of equations with cubic nonlinearity, called the b-family of FORQ/MCH equations, which includes the cubic Camassa–Holm equation (also called Fokas–Olver–Rosenau–Qiao equation) as a special case. We first study the local well-posedness for the Cauchy problem of the equation, and then make good use of fine structure of the equation, we derive the precise blow-up scenario and a new blow-up result with respect to initial data. Finally, peakon solutions are derived.     

On the persistence and blow up for the generalized two-component Dullin–Gottwald–Holm system

Considered herein is the persistence property of the solutions to the generalized two-component integrable Dullin–Gottwald–Holm system, which was derived from the Euler equation with nonzero constant vorticity in shallow water waves moving over a linear shear flow. Firstly, the persistence properties of the system are investigated in weighted $$L^p$$-spaces for a large class of moderate weights. Then, we establish the new local-in-space blow-up results simplifying and extending earlier blow-up criterion for this system.     

Self-similar solutions and blow-up phenomena for a two-component shallow water system

In this article, we consider a two-component nonlinear shallow water system, which includes the famous 2-component Camassa-Holm and Degasperis-Procesi equations as special cases. The local well-posedess for this equations is established. Some sufficient conditions for blow-up of the solutions in finite time are given. Moreover, by separation method, the self-similar solutions for the nonlinear shallow water equations are obtained, and which local or global behavior can be determined by the corresponding Emden equation.     

On Exact Multidimensional Solutions of a Nonlinear System of Reaction–Diffusion Equations

We study a nonlinear reaction–diffusion system modeled by a system of two parabolic-type equations with power-law nonlinearities. Such systems describe the processes of nonlinear diffusion in reacting two-component media. We construct multiparameter families of exact solutions and distinguish the cases of blow-up solutions and exact solutions periodic in time and anisotropic in spatial variables that can be represented in elementary functions.     

Blow-up phenomena and local well-posedness for a generalized Camassa–Holm equation in the critical Besov space

In this paper we mainly study the Cauchy problem for a generalized Camassa–Holm equation in a critical Besov space. First, by using the Littlewood–Paley decomposition, transport equations theory, logarithmic interpolation inequalities and Osgood’s lemma, we establish the local well-posedness for the Cauchy problem of the equation in the critical Besov space $$B^{\frac{1}{2}}_{2,1}$$. Next we derive a new blow-up criterion for strong solutions to the equation. Then we give a global existence result for strong solutions to the equation. Finally, we present two new blow-up results and the exact blow-up rate for strong solutions to the equation by making use of the conservation law and the obtained blow-up criterion.     

Global existence and blow-up phenomena for a weakly dissipative 2-component Camassa–Holm system

We study the Cauchy problem of a weakly dissipative 2-component Camassa–Holm system. We first establish local well-posedness for a weakly dissipative 2-component Camassa–Holm system. We then present a global existence result for strong solutions to the system. We finally obtain several blow-up results and the blow-up rate of strong solutions to the system.     

具零阶耗散的双成分Camassa-Holm方程的整体解和爆破现象

本文研究了具零阶耗散的双成分Camassa-Holm方程的Cauchy问题.由Kato定理得到局部适定性的结果,然后研究了解的整体存在性和爆破现象.     

Blow-up Dynamics of L~2 Solutions for the Davey–Stewartson System

We study the blow-up solutions for the Davey–Stewartson system(D–S system, for short)in L2x(R2). First, we give the nonlinear profile decomposition of solutions for the D–S system. Then, we prove the existence of minimal mass blow-up solutions. Finally, by using the characteristic of minimal mass blow-up solutions, we obtain the limiting profile and a precisely mass concentration of L2 blow-up solutions for the D–S system.     

On the wave-breaking phenomena for the periodic two-component Dullin–Gottwald–Holm system

Considered herein is the well-posedness problem of the periodic two-component Dullin–Gottwald–Holm (DGH) system on the circle, which can be derived from Euler?s equation with constant vorticity in shallow water waves moving over a linear shear flow. The result of blow-up solutions for certain initial profiles in a manner which corresponds to wave-breaking is established.