CPAA

We establish the local well-posedness for the periodic generalized
Camassa-Holm equation. We also give the precise blow-up scenario
and prove that the equation has smooth solutions that blow up in
finite time.

DCDS

In this paper, we mainly study the Cauchy problem of an integrable dispersive Hunter-Saxton equation in periodic domain. Firstly, we establish local well-posedness of the Cauchy problem of the equation in $H^s (\mathbb{S}), s > \frac{3}{2},$ by applying the Kato method. Secondly, by using some conservative quantities, we give a precise blow-up criterion and a blow-up result of strong solutions to the equation. Finally, based on a sign-preserve property, we transform the original equation into the sinh-Gordon equation. By using the travelling wave solutions of the sinh-Gordon equation and a period stretch between these two equations, we get the travelling wave solutions of the original equation.

DCDS

In this paper, we mainly consider the Cauchy problem for a generalized two-component shallow water wave system with fractional higher-order inertia operators: $m=(1-\partial_x^2)^su, s>1$. By Littlewood-Paley theory and transport equation theory, we first establish the local well-posedness of the generalized b-equation with fractional higher-order inertia operators which is the subsystem of the generalized two-component water wave system. Then we prove the local well-posedness of the generalized two-component water wave system with fractional higher-order inertia operators. Next, we present the blow-up criteria for these systems. Moreover, we obtain some global existence results for these systems.

DCDS

This paper is concerned with a four-component Camassa-Holm type system proposed in [37], where its bi-Hamiltonian structure and infinitely many conserved quantities were constructed. In the paper, we first establish the local well-posedness for the system. Then we present several global existence and blow-up results for two integrable two-component subsystems.

DCDS

We establish the local well-posedness for a
recently derived model that combines the linear dispersion of
Korteweg-de Veris equation with the nonlinear/nonlocal dispersion
of the Camassa-Holm equation, and we prove that the equation has
solutions that exist for indefinite times as well as solutions
that blow up in finite 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.

DCDS

In this paper we mainly study the Cauchy problem for a generalized Camassa-Holm equation. First, by using the Littlewood-Paley decomposition and transport equations theory, we establish the local well-posedness for the Cauchy problem of the equation in Besov spaces. Then we give a blow-up criterion for the Cauchy problem of the equation. we present a blow-up result and the exact blow-up rate of strong solutions to the equation by making use of the conservation law and the obtained blow-up criterion. Finally, we verify that the system possesses peakon solutions.

DCDS

We first establish the local existence and uniqueness of strong solutions for the Cauchy problem of a
generalized Camassa-Holm equation in nonhomogeneous Besov spaces by using the Littlewood-Paley theory.
Then, we prove that the solution depends continuously on the initial data in the corresponding Besov space. Finally, we derive a blow-up criterion and present a blow-up result and a blow-up rate of the blow-up solutions to the equation.

DCDS

This paper is concerned with the problem of well-posedness for a modified two-component Camassa-Holm
system in Besov spaces with the critical index $s=\frac 3 2$.

DCDS

In this paper we mainly investigate the Cauchy problem of a three-component Camassa-Holm system. By using Littlewood-Paley theory and transport equations theory, we establish the local well-posedness of the system in the critical Besov space. Moreover, we obtain some weighted $L^p$ estimates of strong solutions to the system. By taking suitable weighted functions, we can get the persistence properties of strong solutions on exponential, algebraic and logarithmic decay rates, respectively.

DCDS

This paper is concerned with the the initial value problem for higher dimensional Camassa-Holm equations.
Firstly, the local well-posedness for this equations in both supercritical and critical Besov spaces are established.
Then two blow-up criterions of strong solutions to the equations are derived.
Finally, the analyticity of its solutions is proved in both variables, globally in space and locally in time.