Well-posedness and blow-up for a modified two-component Camassa–Holm equation

In this article, we consider a newly modified two-component Camassa–Holm equation. First, we establish the local well-posedness result, then we present a precise blow-up scenario. Afterwards, we derive a new conservation law, by which and the precise blow-up scenario we prove three blow-up results and a blow-up rate estimate result.     

Non-uniform dependence for higher dimensional Camassa–Holm equations in Besov spaces

In this paper, we investigate the dependence on initial data of solutions to higher dimensional Camassa–Holm equations. We show that the data-to-solution map is not uniformly continuous dependence in Besov spaces.     

Solitons and peakons of a nonautonomous Camassa–Holm equation

The Camassa–Holm equation can be used in fluids and other fields. Under investigation in this paper, the bilinear form, implicit soliton solution and multi-peakon solution of the generalized nonautonomous Camassa–Holm equation under constraints are derived. Based on these, time varying influence factors of solution amplitude, velocity and background are discussed, which are caused by inhomogeneity of boundaries and media. Furthermore, the phenomena of nonlinear tunnelling, soliton collision and split are constructed to show the characteristic of nonautonomous solitons and peakons in the propagation.     

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.     

A note on regularity criteria for the viscous Camassa–Holm equations in multiplier spaces

In this note, regularity criteria for the viscous Camassa–Holm equations are established in multiplier spaces; these improve on previous results.     

Well-posedness of fractional Ginzburg–Landau equation in Sobolev spaces

This article is concerned with real fractional Ginzburg–Landau equation. Existence and uniqueness of local and global mild solution for both whole space case and flat torus case are obtained by contraction semigroup method, and Gevrey regularity of mild solution for flat torus case is discussed.     

Global attractor for the viscous modified two-component Camassa–Holm equation

We obtain the existence of global attractor for the Cauchy problem of a viscous modified two-component Camassa–Holm equation. The existence of global strong solutions is obtained using Kato’s theory. The key elements in our analysis are the uniform Gronwall lemma and some estimates of the solutions.     

Oscillation-induced blow-up to the modified Camassa–Holm equation with linear dispersion

In this paper, we provide a blow-up mechanism to the modified Camassa–Holm equation with varying linear dispersion. We first consider the case when linear dispersion is absent and derive a finite-time blow-up result with an initial data having a region of mild oscillation. A key feature of the analysis is the development of the Burgers-type inequalities with focusing property on characteristics, which can be deduced from tracing the ratio between solution and its gradient. Using the continuity and monotonicity of the solutions, we then extend this blow-up criterion to the case of negative linear dispersion, and determine that the finite time blow-up can still occur if the initial momentum density is bounded below by the magnitude of the linear dispersion and the initial datum has a local mild-oscillation region. Finally, we demonstrate that in the case of non-negative linear dispersion the formation of singularities can be induced by an initial datum with a sufficiently steep profile. In contrast to the Camassa–Holm equation with linear dispersion, the effect of linear dispersion of the modified Camassa–Holm equation on the blow-up phenomena is rather delicate.     

Optimal control of the viscous Camassa–Holm equation

This paper studies the problem of optimal control of the viscous Camassa–Holm equation. The existence and uniqueness of weak solution to the viscous Camassa–Holm equation are proved in a short interval. According to variational method, optimal control theories and distributed parameter system control theories, we can deduce that the norm of solution is related to the control item and initial value in the special Hilbert space. The optimal control of the viscous Camassa–Holm equation under boundary condition is given and the existence of optimal solution to the viscous Camassa–Holm equation is proved.     

Bifurcations of travelling wave solutions for a class of nonlinear fourth order variant of a generalized Camassa–Holm equation

In this paper, by using the bifurcation theory of dynamical systems for a class of nonlinear fourth order variant of a generalized Camassa–Holm equation, the existence of solitary wave solutions, breaking bounded wave solutions, compacton solutions and non-smooth periodic wave solutions are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. Some exact explicit parametric representations of the above waves are determined.     

Blow-up criteria of solutions to a modified two-component Camassa–Holm system

In this paper, we consider a modified two-component Camassa–Holm (MCH2) system which arises in shallow water theory. We analyze the wave breaking mechanism by establishing some new blow-up criteria for this system formulated either on the line or with space-periodic initial condition.     

Wave breaking for a modified two-component Camassa–Holm system

In this paper, we establish sufficient conditions on the initial data to guarantee blow-up phenomenon for the modified two-component Camassa–Holm (MCH2) system.     

Well-posedness and blowup phenomena for a cross-coupled Camassa–Holm equation with waltzing peakons and compacton pairs

This paper deals with the Cauchy problem for a cross-coupled Camassa–Holm equation $$m_t=-(vm)_x-mv_x, n_t=-(un)_x-nu_x,$$ where \({n\doteq v-v_{xx}}\) , \({m\doteq u-u_{xx}+\omega}\) with a constant ω. The local well-posedness of solutions for the Cauchy problem of the cross-coupled Camassa–Holm equation in Sobolev space \({H^s(\mathbb{R})}\) with s > 5/2 is established. Under some assumptions, the existence and uniqueness of the global solutions to the equation are shown, and the blowup scenario of the solutions to the equation is also obtained.     

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.     

Optimal control of the viscous generalized Camassa–Holm equation

In this paper, we study the optimal control problem for the viscous generalized Camassa–Holm equation. We deduce the existence and uniqueness of weak solution to the viscous generalized Camassa–Holm equation in a short interval by using Galerkin method. Then, by using optimal control theories and distributed parameter system control theories, the optimal control of the viscous generalized Camassa–Holm equation under boundary condition is given and the existence of optimal solution to the viscous generalized Camassa–Holm equation is proved.