On initial-boundary value problems in bounded and unbounded domains for a class of nonlinear hyperbolic equations of the third order

For the nonlinear hyperbolic equation

u(2,1)=f(x,t,u,u(1,0),u(2,0),u(0,1),u(1,1))     

On Nonlinear Hyperbolic Functional Differential Equations

In this paper, existence of weak solutions of second order evolution equations is proved and some properties of the solutions are shown. The results are applied to higher order nonlinear hyperbolic functional differential equations.     

Optimization for nonlinear hyperbolic equations without the uniqueness theorem for a solution of the boundary-value problem

We consider the optimal control problem for a system governed by a nonlinear hyperbolic equation without any constraints on the parameter of nonlinearity. No uniqueness theorem is established for a solution to this problem. The control-state mapping of this system is not Gateaux differentiable. We study an approximate solution of the optimal control problem by means of the penalty method.     

Hyperbolic–parabolic singular perturbation for quasilinear equations of Kirchhoff type with weak dissipation

We consider a hyperbolic–parabolic singular perturbation problem for a quasilinear hyperbolic equation of Kirchhoff type with dissipation weak in time. The purpose of this paper is to give time‐decay convergence estimates of the difference between the solutions of the hyperbolic equation above and those of the corresponding parabolic equation, together with the unique existence of the global solutions of the hyperbolic equation above. Copyright © 2009 John Wiley & Sons, Ltd.     

Oscillatory properties for semilinear degenerate hyperbolic equations of second order

We consider three kind of oscillatory properties of the solutions to semilinear degenerate hyperbolic equations. Several sufficient conditions for the oscillation or non-oscillation are presented. In particular, they give us the positivity of the solutions for semilinear hyperbolic equations degenerating at initial point in one space dimension. Moreover we establish a few oscillatory conditions for the solutions of the mixed problem reduced to in one space dimension.     

An approximation of stochastic hyperbolic equations: case with Wiener process

In the present paper, the two‐step difference scheme for the Cauchy problem for the stochastic hyperbolic equation is presented. The convergence estimate for the solution of the difference scheme is established. In applications, the convergence estimates for the solution of difference schemes for the numerical solution of four problems for hyperbolic equations are obtained. The theoretical statements for the solution of this difference scheme are supported by the results of the numerical experiment. Copyright © 2012 John Wiley & Sons, Ltd.     

Oscillation of solutions of impulsive vector hyperbolic differential equations with delays

In this article, we investigate a class of impulsive vector hyperbolic differential equation with delays. Several sufficient conditions are established for H-oscillation of solutions of such systems subject to the Neumann boundary condition by employing certain second-order impulsive differential inequality, where H is a unit vector in ? ^M . The main results are illustrated by two examples.     

常系数线性差分方程的微分解法

用微分方法 ,解常系数线性差分方程     

Nonnegative solutions of the characteristic initial value problem for linear partial functional–differential equations of hyperbolic type

On the rectangle , the problem on the existence and uniqueness of a nonnegative solution of the characteristic initial value problem for the equation

is considered, where is a linear bounded operator and .     

具有Lipschitz连续系数的Kirchhoff方程

作者考虑了一个非局部项仅为Ｌｉｐｓｃｈｉｔｚ连续函数的非齐次Ｋｉｒｃｈｈｏｆ方程的Ｃａｕｃｈｙ问题．在初值和右端项“小”的条件下，我们获得了此问题整体解的存在唯一性．     

变系数高阶中立型微分方程的振动性

考虑变系数高阶中立型微分方程(NDDE)d~n/(dt~n)[y(t)+p(t)y(t-τ)]+sum from n=1 to ∞q~i(t)y(t-σ_i)=0 (1)其中p(t)、g_i(t)都是区间[T,∞)上连续的实值函数.p(t)有界,q_i(t)≥0(i=1,2,···,m)且至少有一个q_i(t)最终大于某一任意小的正数.τ≥0,σ_i≥0.m≥1,n≥1均为正整数. 本文研究了方程(1)在p(t)≥一1及p(t)≤-1等情况下解的渐近性和振动性,获得了一系列使解振动的充分条件.特别,p(t)有时可以是变号函数.     

On error estimates in the Galerkin method for hyperbolic equations

We consider the Cauchy problem in a Hilbert space for a second-order abstract quasilinear hyperbolic equation with variable operator coefficients and nonsmooth (but Bochner integrable) free term. For this problem, we establish an a priori energy error estimate for the semidiscrete Galerkin method with an arbitrary choice of projection subspaces. Also, we establish some results on existence and uniqueness of an exact weak solution. We give an explicit error estimate for the finite element method and the Galerkin method in Mikhlin form.     

Stability of nonautonomous difference equations with several delays

We consider the possibility to construct efficient stability criteria for solutions to difference equations with variable coefficients. We prove that one can associate a difference equation with a certain functional differential equation, whose solution has the same asymptotic behavior. We adduce examples, demonstrating the essential character of conditions of the obtained theorems and the exactness of the constant 3/2 which defines the boundary of the stability domain.     

On reduction of some classes of partial differential equations to equations with fewer variables and exact solutions

We establish a connection between the fundamental solutions to some classes of linear nonstationary partial differential equations and the fundamental solutions to other nonstationary equations with fewer variables. In particular, reduction enables us to obtain exact formulas for the fundamental solutions of some spatial nonstationary equations of mathematical physics (for example, the Kadomtsev-Petviashvili equation, the Kelvin-Voigt equation, etc.) from the available fundamental solutions to one-dimensional stationary equations.     

ANALYSIS OF STRONG SINGULARITIES FOR NONLINEAR HYPERBOLIC EQUATIONS

In this paper we review our some results about the strongly singular (discontinuous,measure, or delta function) problems for nonlinear hyperbolic equations.     

Global solvability for the Kirchhoff equations in exterior domains of dimension larger than three

We consider the unique global solvability of initial (boundary) value problem for the Kirchhoff equations in exterior domains or in the whole Euclidean space for dimension larger than three. The following sufficient condition is known: initial data is sufficiently small in some weighted Sobolev spaces for the whole space case; the generalized Fourier transform of the initial data is sufficiently small in some weighted Sobolev spaces for the exterior domain case. The purpose of this paper is to give sufficient conditions on the usual Sobolev norm of the initial data, by showing that the global solvability for this equation follows from a time decay estimate of the solution of the linear wave equation. Copyright © 2004 John Wiley & Sons, Ltd.     

Large time behavior of solutions to degenerate parabolic equations with weights

We study the degenerate parabolic equation t_∂u=a(δ(x))upΔu−g(u) in Ω×(0,∞), where Ω⊂RN (N?1) is a smooth bounded domain, p?1, δ(x)=dist(x,∂Ω) and a is a continuous nondecreasing function such that a(0)=0. Under some suitable assumptions on a and g we prove the existence and the uniqueness of a classical solution and we study its asymptotic behavior as t→∞.     

Lamé equations with algebraic solutions

In this paper we study Lamé equations L_n,By=0 in so-called algebraic form, having only algebraic functions as solution. In particular we provide a complete list of all finite groups that occur as the monodromy groups, together with a list of examples of such equations. We show that the set of such Lamé equations with is countable, up to scaling of the equation. This result follows from the general statement that the set of equivalent second-order equations, having algebraic solutions and all of whose integer local exponent differences are 1, is countable.