On a Nonlocal Problem with Integral Conditions for the System of Hyperbolic Equations

A nonlocal problem with integral conditions is considered for the system of partial differential equations of the hyperbolic type in a rectangular domain. Sufficient conditions are established for the existence of the unique classical solution of the studied problem in terms of initial data. An algorithm is proposed for finding a sequence of approximate solutions convergent to the exact solution of the problem. Special cases of the problem at hand are considered as an application of the results obtained.     

On a Nonlocal BVP with Nonlinear Boundary Conditions

We consider the existence of at least one positive solution of the problem ${-y''(t)=f(t,y(t)), y(0)=H_1(\varphi(y))+\int_{E}H_2(s,y(s))\,ds, y(1)=0}$ , where ${y(0)=H_1(\varphi(y))+\int_{E}H_2(s,y(s))\,ds}$ represents a nonlinear, nonlocal boundary condition. We show by imposing some relatively mild structural conditions on f, H ₁, H ₂, and ${\varphi}$ that this problem admits at least one positive solution. Finally, our results generalize and improve existing results, and we give a specific example illustrating these generalizations and improvements.     

On Exact Multidimensional Solutions to a Nonlinear System of Fourth-Order Hyperbolic Equations

Journal of Applied and Industrial Mathematics - We study the system of two fourth-order nonlinear hyperbolic partial differential equations. The right-hand sides of the equations contain double...     

Profile of Blow-up Solution to Hyperbolic System with Nonlocal Term

This paper is concerned with a nonlocal hyperbolic system as follows utt = △u ＋ （∫Ωvdx ）^p for x∈R^N,t〉0 ,utt = △u ＋ （∫Ωvdx ）^q for x∈R^N,t〉0 ,u（x,0）=u0（x）,ut（x,0）=u01（x） for x∈R^N,u（x,0）=u0（x）,ut（x,0）=u01（x） for x∈R^N, where 1≤ N ≤3, p ≥1, q ≥ 1 and pq 〉 1. Here the initial values are compactly supported and Ω belong to R^N is a bounded open region. The blow-up curve, blow-up rate and profile of the solution are discussed.     

Nonlinear Models of Suspension Bridges: Discussion of the Results

In this paper we present several nonlinear models of suspension bridges; most of them have been introduced by Lazer and McKenna. We discuss some results which were obtained by the authors and other mathematicians for the boundary value problems and initial boundary value problems. Our intention is to point out the character of these results and to show which mathematical methods were used to prove them instead of giving precise proofs and statements.     

可化约拟线性双曲组带一类非局部边界条件的精确边界能控性

先证明了一类非线性积分方程解的存在唯一性,利用此结果,建立了可化约拟线性双曲组带一类非局部边界条件的单侧精确边界能控性.     

Critical Nonlinear Nonlocal Equations on a Half-Line

We study nonlinear nonlocal equations on a half-line in the critical case

Classical Solutions of Hyperbolic Equations with Nonlocal Potentials

Doklady Mathematics - A three-parameter family of global solutions for a two-dimensional hyperbolic differential-difference equation with a nonlocal potential is constructed. A theorem that the...     

一类非线性耦合双曲型方程组的研究

研究了一类非线性耦合双曲型方程组初边值问题,讨论了初边值条件以及非线性项中指数满足的条件,利用Galerkin方法和紧致性的结果,以及椭圆型方程解的正则性定理,经过严格的推导证明了该问题解的存在唯一性的若干结果.     

关于一类非线性双曲方程组的二维Cauchy问题

本文通过引进新的广义解定义，对一类非线性双曲方程组的二维Ｃａｕｃｈｙ问题，证明了解的存在唯一性．并且，解可能含δ波．     

On the Decay of Solution of Some Nonlinear Hyperbolic Equation

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎａｎｄＲｅｓｕｌｔＩｎｔｈｉｓａｒｔｉｃｌｅｗｅａｒｅｃｏｎｃｅｒｎｅｄｗｉｔｈｔｈｅｄｅｃａｙｏｆｇｌｏｂａｌｓｏｌｕｔｉｏｎｏｆｔｈｅｉｎｉｔｉａｌ－ｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｆｏｒｔｈｅｆｏｌｌｏｗｉｎｇｎｏｎｌｉｎｅａｒｈｙｐｅｒｂｏｌｉｃｅｑｕａｔｉｏｎｕｔｔ＋Ａｕ＋｜ｕｔ｜αｕｔ＝ｆ（ｘ,ｔ）　　　　　ｉｎΩ×Ｒ＋,（１）ｕ（ｘ,０）＝ｕ０（ｘ）,ｕｔ（ｘ,０）＝ｕ１（ｘ）　　ｘ∈Ω,（２）ｕ（ｘ,ｔ）＝０　　　　　　　　　　　　（ｘ,ｔ…     

一非线性双曲型方程解的爆破

本文应用压缩映射原理证明一非线性双曲型方程的初边值问题存在唯一局部广义解，并给出此问题的广义解爆破的充分条件。     

Coupled String-Beam Equations as a Model of Suspension Bridges

We consider nonlinearly coupled string-beam equations modelling time-periodic oscillations in suspension bridges. We prove the existence of a unique solution under suitable assumptions on certain parameters of the bridge.     

多维非线性脉冲波的单波“干扰”

利用非线性几何光学讨论3×3的高维半线性双曲偏微分方程在仅有一个初始脉冲波的条件下,系统会产生多个脉冲波的现象,从而从数学的角度论证了单个脉冲波的传播与"干扰"特性.