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This paper is concerned with a new integrable multi-component Camassa–Holm system, which has been proven to be integrable and has peakon solutions. This system includes many one-component and two-component Camassa–Holm type systems as special cases. In this paper, we first establish the local well-posedness and a continuation criterion for the system, then we present several global existence and blow-up results for two important integrable two-component subsystems. Our obtained results cover and improve considerably recent results in Gui et al. (2013), Yan et al. (2015). [ABSTRACT FROM AUTHOR]

In the paper (Bui and Reissig, 2015) we explained that it is reasonable to divide the study of global existence of small data solutions to semi-linear classical damped wave models with time-dependent speed of propagation, time-dependent dissipation and power nonlinearity into two cases. The super-exponential case was treated in Bui and Reissig (2015). The present paper is devoted to the sub-exponential case. Both cases arise from a different influence of the interplay of time-dependent coefficients on the critical exponent. Here we only sketch differences to the approach for the super-exponential case. These differences appear in handling the nonlinearity. The corresponding Matsumura type estimates for a family of linear Cauchy problems depending on a parameter coincide in both cases (see Bui and Reissig (2014, 2015)). [ABSTRACT FROM AUTHOR]

As we all know, the conservation laws take an important part in deriving wave breaking for Degasperis–Procesi (DP) type equation. In the article, we give a new method to study singularities in finite time for a class of DP equations with high nonlinear terms. The paper tells us that the result of wave breaking can be established without the conservation law. [ABSTRACT FROM AUTHOR]

In this paper, we prove the uniqueness and generic regularity of the energy conservative solution for a system of wave equations modeling nematic liquid crystal. [ABSTRACT FROM AUTHOR]

In the significant work of [6] , Alinhac proved the global existence of small solutions for 2D quasilinear wave equations under the null conditions. The proof heavily relies on the fact that the initial data have compact support [23] . Whether this constraint can be removed or not is still unclear. In this paper, for fully nonlinear wave equations under the null conditions, we prove the global well-posedness for small initial data without compact support. Moreover, we apply our result to a class of quasilinear wave equations. [ABSTRACT FROM AUTHOR]

By applying the pullback property of attractor, the paper obtains the continuity and the convergent rate p = 1 2 of the attractor for a singularly perturbed wave equation without help of the hyperbolic property of stationary points. [ABSTRACT FROM AUTHOR]

We study nonlocal equations from the area of peridynamics on bounded domains. We present four main results. In our recent paper, we have discovered that, on R, the governing operator in peridynamics, which involves a convolution, is a bounded function of the classical (local) governing operator. Building on this, as main result 1, we construct an abstract convolution operator on bounded domains which is a generalization of the standard convolution based on integrals. The abstract convolution operator is a function of the classical operator, defined by a Hilbert basis available due to the purely discrete spectrum of the latter. As governing operator of the nonlocal equation we use a function of the classical operator, this allows us to incorporate local boundary conditions into nonlocal theories. As main result 2, we prove that the solution operator can be uniquely decomposed into a Hilbert—Schmidt operator and a multiple of the identity operator. As main result 3, we prove that Hilbert—Schmidt operators provide a smoothing of the input data in the sense a square integrable function is mapped into a function that is smooth up to boundary of the domain. As main result 4, for the homogeneous nonlocal wave equation, we prove that continuity is preserved by time evolution. Namely, the solution is discontinuous if and only if the initial data is discontinuous. As a consequence, discontinuities remain stationary. [ABSTRACT FROM AUTHOR]

In this paper, we present a new family of locally one-dimensional (LOD) schemes with fourth-order accuracy in both space and time for the three-dimensional (3D) acoustic wave equation. It is well-known that high order explicit schemes offer efficient time steppings but with restrictive CFL conditions; implicit discretization gives unconditional stable schemes but with inefficient time steppings. Our LOD schemes can be seen as a compromise between explicit and implicit schemes, in the way that our schemes have more relax CFL conditions compared with traditional explicit schemes and only require solutions of tridiagonal systems of linear algebraic equations, which is more efficient than implicit schemes. Moreover, the new scheme is four-layer in time and three-layer in space, so that boundary conditions can be imposed in the classical way. The computations of the initial conditions for the three intermediate time layers are explicitly constructed. Furthermore, the stability condition is derived and the restriction for the time step is given explicitly. Finally, numerical examples are completed to show the performance of our new method. [ABSTRACT FROM AUTHOR]

This paper is concerned with the periodic Cauchy problem for a generalized Camassa–Holm integrable equation, which can be viewed as a generalization to both the Camassa–Holm (CH) and modified Camassa–Holm (mCH) equations. We mainly make a detailed presentation on the effects of varying the CH and mCH nonlocal nonlinearities on the non-uniform dependence and Hölder continuity of the solution map. Using a Galerkin-type approximation method, we first establish the local well-posedness result in Sobolev spaces H s , s > 5 2 , with continuous dependence on the initial data. Then we prove that this dependence is sharp by showing that the data-to-solution map is not uniformly continuous, which is based on well-posedness estimates and the method of approximate solutions. Furthermore, we demonstrate that the solution map is Hölder continuous in the H σ topology with 0 ≤ σ < s . [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the long time behavior of solutions to the initial boundary value problem of critical semilinear wave equations with small data in exterior domain in high dimensions ( n ≥ 5 ). We prove that solutions cannot exist globally in time. The novelty is that we first use a cutoff function to transform the exterior problem to a Cauchy problem. Then we divide the time into two intervals, in one time interval we establish an energy estimate to control the error terms while in the other interval the asymptotic behavior of two test functions is used. Furthermore, we obtain the upper bound of the lifespan by following the idea of Zhou and Han (2014). [ABSTRACT FROM AUTHOR]

This paper develops a new explicit semi-analytical approach to solving PDEs based on the variation-of-constants formula, or the equivalent integration equation. These new schemes avoid the discretization of spatial derivatives. Therefore, the accuracy of the semi-analytical explicit scheme depends entirely on time integration. We first establish a semi-analytical formula for wave partial differential equations and show the consistency of the semi-analytical formula with different boundary conditions. We then present semi-analytical explicit RKN-type integrators for solving wave PDEs. This methodology is also extended to dealing with first-order hyperbolic PDEs and parabolic PDEs, and for both of them the corresponding semi-analytical explicit RK-type integrators are derived as well. Numerical simulations are implemented and the results show the effectiveness of the new semi-analytical explicit schemes. Meanwhile, an operator-variation-of-constants formula with applications is introduced for high-dimensional nonlinear wave equations before presenting the idea of semi-analytical explicit RKN integrators. [ABSTRACT FROM AUTHOR]

In this paper we analyze time marching schemes for the wave equation in mixed form. The problem is discretized in space using stabilized finite elements. On the one hand, stability and convergence analyses of the fully discrete numerical schemes are presented using different time integration schemes and appropriate functional settings. On the other hand, we use Fourier techniques (also known as von Neumann analysis) in order to analyze stability, dispersion and dissipation. Numerical convergence tests are presented for various time integration schemes, polynomial interpolations (for the spatial discretization), stabilization methods, and variational forms. To analyze the behavior of the different schemes considered, a 1D wave propagation problem is solved. [ABSTRACT FROM AUTHOR]

Wave breaking of the water wave is important and interesting to physicists and mathematicians. In this paper, based on the conservation laws and the blow-up scenario, some new blow-up phenomena are derived for the Camassa–Holm equation and Degasperis–Procesi equation, which have attracted much attention due to their structure. [ABSTRACT FROM AUTHOR]

In this paper we study the initial boundary value problem of critical semilinear wave equation with small initial data in 3-D. Blow up result will be established no matter how small the data are. We find a radial and positive solution of the corresponding linear problem, then our blow up result follows from the idea of Zhou and Han (2014). What is more, we get the upper bound of the lifespan estimate. [ABSTRACT FROM AUTHOR]

We consider semi-linear classical damped wave models with time-dependent speed of propagation and time-dependent dissipation. The right-hand side is a power nonlinearity which can be interpreted only as a source term. We are interested in the interplay of time-dependent coefficients on the global existence of small data solutions. The considerations are divided into two cases depending on the behavior of the propagation speed: the super-exponential and the sub-exponential case. In this paper we study the super-exponential case. One main tool of our approach is the use of Matsumura type estimates for a family of Cauchy problems depending on a parameter. [ABSTRACT FROM AUTHOR]

In this paper, we consider the higher-order asymptotic attraction of pullback attractors for a reaction–diffusion equation with domains expanding in time. Firstly, to make the test functions meaningful, a maximum principle is proved for the corresponding variational solution; then, we establish some higher-order integrability results about the difference of variational solutions, and finally we prove the main result that the known ( L 2 , L 2 ) pullback D λ -attractor can attract indeed the D λ class in L q -norm for any q ∈ [ 2 , + ∞ ) . [ABSTRACT FROM AUTHOR]

This paper is concerned with the Cauchy problem for a two-component b-family system. We prove that the solution map of the Cauchy problem of the b-family system is not uniformly continuous in H s ( R ) × H s − 1 ( R ) , s > 5 / 2 . [ABSTRACT FROM AUTHOR]

Abstract: In this paper we develop numerical approximations of the wave equation in mixed form supplemented with non-reflecting boundary conditions (NRBCs) of Sommerfeld-type on artificial boundaries for truncated domains. We consider three different variational forms for this problem, depending on the functional space for the solution, in particular, in what refers to the regularity required on artificial boundaries. Then, stabilized finite element methods that can mimic these three functional settings are described. Stability and convergence analyses of these stabilized formulations including the NRBC are presented. Additionally, numerical convergence test are evaluated for various polynomial interpolations, stabilization methods and variational forms. Finally, several benchmark problems are solved to determine the accuracy of these methods in 2D and 3D. [Copyright &y& Elsevier]