Initial Boundary Value Problems of the Camassa–Holm Equation

In this paper we study initial boundary value problems of the Camassa–Holm equation on the half line and on a compact interval. Using rigorously the conservation of symmetry, it is possible to convert these boundary value problems into Cauchy problems for the Camassa–Holm equation on the line and on the circle, respectively. Applying thus known results for the latter equations we first obtain the local well-posedness of the initial boundary value problems under consideration. Then we present some blow-up and global existence results for strong solutions. Finally we investigate global and local weak solutions for the equation on the half line and on a compact interval, respectively. An interesting result of our analysis shows that the Camassa–Holm equation on a compact interval possesses no nontrivial global classical solutions.     

n阶非线性常微分方程非线性k点边值问题解的存在性

利用格林函数和上下解方法讨论了n阶非线性常微分方程之几类具有非线性k点边界条件的边值问题的存在性。     

Initial boundary value problems for the compressible viscoelastic fluid

This paper is concerned with the initial boundary value problems of the system modeling the compressible viscoelastic fluid of Oldroyd type. The global in time solution is proved to exist uniquely near the equilibrium state in H².     

Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems

We consider initial value/boundary value problems for fractional diffusion-wave equation: , where 0<α?2, where L is a symmetric uniformly elliptic operator with t-independent smooth coefficients. First we establish the unique existence of the weak solution and the asymptotic behavior as the time t goes to ∞ and the proofs are based on the eigenfunction expansions. Second for α∈(0,1), we apply the eigenfunction expansions and prove (i) stability in the backward problem in time, (ii) the uniqueness in determining an initial value and (iii) the uniqueness of solution by the decay rate as t→∞, (iv) stability in an inverse source problem of determining t-dependent factor in the source by observation at one point over (0,T).     

The initial boundary value problem for quasi-linear wave equation with viscous damping

In this paper, the existence and uniqueness of the local generalized solution and the local classical solution for the initial boundary value problem of the quasi-linear wave equation with viscous damping are proved. The nonexistence of the global solution for this problem is discussed by an ordinary differential inequality. Finally, an example is given.     

Moving boundary value problems for the wave equation

We study certain boundary value problems for the one-dimensional wave equation posed in a time-dependent domain. The approach we propose is based on a general transform method for solving boundary value problems for integrable nonlinear PDE in two variables, that has been applied extensively to the study of linear parabolic and elliptic equations. Here we analyse the wave equation as a simple illustrative example to discuss the particular features of this method in the context of linear hyperbolic PDEs, which have not been studied before in this framework.     

Two-point and three-point boundary value problems for n th-order nonlinear differential equations

In this article, we are concerned with existence and uniqueness of solutions of four kinds of two-point boundary value problems for nth-order nonlinear differential equations by “Shooting” method, and studied existence and uniqueness of solutions of a kind of three-point boundary value problems for nth-order nonlinear differential equations by “Matching” method.     

Initial boundary value problem and asymptotic stabilization of the Camassa-Holm equation on an interval

We investigate the nonhomogeneous initial boundary value problem for the Camassa-Holm equation on an interval. We provide a local in time existence theorem and a weak-strong uniqueness result. Next we establish a result on the global asymptotic stabilization problem by means of a boundary feedback law.     

Global existence and blow-up of solutions for a system of nonlinear viscoelastic wave equations with damping and source

In this paper we investigate the global existence and finite time blow-up of solutions to the system of nonlinear viscoelastic wave equations

in Ω×(0,T) with initial and Dirichlet boundary conditions, where Ω is a bounded domain in . Under suitable assumptions on the functions g_i(), , the initial data and the parameters in the equations, we establish several results concerning local existence, global existence, uniqueness and finite time blow-up property.     

Nonlinear boundary value problems for discontinuous delayed differential equations

In this paper nonlinear boundary value problems for discontinuous delayed differen- tial equations are considered. Some existence and boundedness results of solutions are obtained via the method of upper and lower solutions, which may be discontinuous. Our analysis can be applied to those phenomena from physics and control theory which have been successfully described by delayed differential equations with discontinuous functions, such as electric, pneu- matic, and hydraulic networks. It is also an important step to precede the design of control signals when finite-time or practical stability control are concerned.     

Global boundary value problems for nonlinear dynamic equations on time scales

We consider nonlinear boundary value problems for dynamic equations on time scales. We study nonlinear dynamic equations subject to global boundary conditions. Criteria are provided for the solvability of such problems. In the case of weak nonlinearities, we also examine the dependence of the solution on parameters.     

Initial boundary value problem for generalized boussinesque type equations with nonlinear source

A comparison principle for solutions of the first initial boundary value problem for the generalized Boussinesque equation with a nonlinear sourceu _t-Δψ(u)-Δu _t+q(u)=0 is established. By using this comparison principle, we prove new existence and nonexistence theorems for solutions of the first initial boundary value problem in the case of power-law functions ψ (ξ) andq (ξ). Translated fromMathematicheskie Zametki, Vol. 65, No. 1, pp. 70–75, January, 1999.     

A nonlinear diffusion system coupled via nonlinear boundary flux

In this paper, we deal with the global existence and nonexistence of solutions to a nonlinear diffusion system coupled via nonlinear boundary flux. By constructing various kinds of sub- and super-solutions and using the basic properties of M-matrix, we give the necessary and sufficient conditions for global existence of nonnegative solutions. The critical curve of Fujita type is conjectured with the aid of some new results, which extend the recent results of Wang et al. [Nonlinear Anal. 71 (2009) 2134-2140] and Li et al. [J. Math. Anal. Appl. 340 (2008) 876-883] to more general equations.     

Nontrivial solutions of singular nonlinear m-point boundary value problems

In this paper, by using the method of topology degree, some existence theorems of nontrivial solutions for singular nonlinear m-point boundary value problems are established. Our nonlinearity may be singular in its dependent variable.     

一类具非线性边值条件的非线性方程的奇摄动问题

研究了一类具非线性边值条件的非线性方程的奇摄动问题,运用合成展开法构造了问题的形式渐近解,并用微分不等式理论证明了所得渐近解的一致有效性.