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))     

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.     

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.     

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 .     

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.     

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.     

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.     

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.     

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→∞.     

Supercritical biharmonic equations with power-type nonlinearity

We study two different versions of a supercritical biharmonic equation with a power-type nonlinearity. First, we focus on the equation Δ² u = |u| ^p-1 u over the whole space , where n > 4 and p > (n + 4)/(n − 4). Assuming that p < p _c, where p _c is a further critical exponent, we show that all regular radial solutions oscillate around an explicit singular radial solution. As it was already known, on the other hand, no such oscillations occur in the remaining case p ≥ p _c. We also study the Dirichlet problem for the equation Δ² u = λ (1 + u) ^p over the unit ball in , where λ > 0 is an eigenvalue parameter, while n > 4 and p > (n + 4)/(n − 4) as before. When it comes to the extremal solution associated to this eigenvalue problem, we show that it is regular as long as p < p _c. Finally, we show that a singular solution exists for some appropriate λ > 0.      

Attractors of dissipative hyperbolic equations with singularly oscillating external forces

We study a uniform attractor $\mathcal{A}^\varepsilon $ for a dissipative wave equation in a bounded domain Ω ? ?ⁿ under the assumption that the external force singularly oscillates in time; more precisely, it is of the form g ₀(x, t)+ ε^?α g ₁ (x, t/ε), x ∈ Ω, t ∈ ?, where α > 0, 0 < ε ≤ 1. In E = H ₀ ¹ × L ₂, this equation has an absorbing set B ^ε estimated as ‖B ^ε‖ _E ≤C ₁+C ₂ε^?α and, therefore, can increase without bound in the norm of E as ε → 0+. Under certain additional constraints on the function g ₁(x, z), x ∈ Ω, z ∈ ?, we prove that, for 0 < α ≤ α ₀, the global attractors $\mathcal{A}^\varepsilon $ of such an equation are bounded in E, i.e., $\parallel \mathcal{A}^\varepsilon \parallel _E \leqslant C_3 $ , 0 < ε ≤ 1. Along with the original equation, we consider a “limiting” wave equation with external force g ₀(x, t) that also has a global attractor $\mathcal{A}^0 $ . For the case in which g ₀(x, t) = g ₀(x) and the global attractor $\mathcal{A}^0 $ of the limiting equation is exponential, it is established that, for 0 < α ≤ α ₀, the Hausdorff distance satisfies the estimate $dist_E (\mathcal{A}^\varepsilon ,\mathcal{A}^0 ) \leqslant C\varepsilon ^{\eta (\alpha )} $ , where η(α) > 0. For η(α) and α ₀, explicit formulas are given. We also study the nonautonomous case in which g ₀ = g ₀(x, t). It is assumed that sufficient conditions are satisfied for which the “limiting” nonautonomous equation has an exponential global attractor. In this case, we obtain upper bounds for the Hausdorff distance of the attractors $\mathcal{A}^\varepsilon $ from $\mathcal{A}^0 $ , similar to those given above.     

Periodic solutions of Duffing equations with semi-quadratic potential and singularity

In this paper, we deal with the existence of periodic solutions of the second order differential equations x^″+g(x)=p(t) with singularity. We prove that the given equation has at least one periodic solution when g(x) has singularity at origin, satisfies

     

一类非线性双曲型发展方程的孤子解

研究了一类非线性发展方程.首先在无扰动情形下,利用待定函数和泛函同伦映射方法得到了非扰动发展方程的孤子精确解和扰动方程的任意次近似行波孤子解.接着引入一个同伦映射,并选取初始近似函数,再用同伦映射理论,依次求出非线性双曲型发展扰动方程孤子解的各次近似解析解.再利用摄动理论举例说明了用该方法得到的近似解析解的有效性和各次近似解的近似度.最后,简述了用同伦映射方法得到的近似解的意义,指出了用上述方法得到的各次近似解具有便于求解、精度高等优点.