The local and global existence of solutions for a generalized Camassa-Holm equation

A fully nonlinear generalization of the Camassa-Holm equation is investigated. Using the pseudoparabolic regularization technique, its local well-posedness in Sobolev space Hs(R) with s3/2 is established via a limiting procedure. Provided that the initial momentum (1-x2)u0 satisfies the sign condition, u0∈Hs(R)(s3/2) and u0∈L1(R),the existence and uniqueness of global solutions for the equation are shown to be true in the space C([0,∞); Hs(R))∩C1([0,∞);Hs-1(R)).     

THE INITIAL BOUNDARY VALUE PROBLEM FOR A CLASS OF NONLINEAR WAVE EQUATIONS WITH DAMPING TERM

The initial-boundary value problem for the four-order nonlinear wave equation with damping term is derived from diverse physical background such as the study of plate and beams and the study of interaction of water waves. The existence of the global weak solutions to this problem is proved by means of the potential well methods.     

The Riccati Equation Method Combined with the Generalized Extended $(G'/G)$-Expansion Method for Solving the Nonlinear KPP Equation

An analytic study of the nonlinear Kolmogorov-Petrovskii-Piskunov (KPP) equation is presented in this paper. The Riccati equation method combined with the generalized extended $(G''/G)$-expansion method is an interesting approach to find more general exact solutions of the nonlinear evolution equations in mathematical physics. We obtain the traveling wave solutions involving parameters, which are expressed by the hyperbolic and trigonometric function solutions. When the parameters are taken as special values, the solitary and periodic wave solutions are given. Comparison of our new results in this paper with the well-known results are given.     

EXPONENTIAL DECAY FOR A NONLINEAR VISCOELASTIC EQUATION WITH SINGULAR KERNELS

The nonlinear viscoelastic wave equation |ut|ρ utt u utt + t 0 g(t s) u(s)ds + |u|p u = 0,in a bounded domain with initial conditions and Dirichlet boundary conditions is considered.We prove that,for a class of kernels g which is singular at zero,the exponential decay rate of the solution energy.The result is obtained by introducing an appropriate Lyapounov functional and using energy method similar to the work of Tatar in 2009.This work improves earlier results.     

具位势项的快扩散方程的第二临界指标

This paper considers a fast diffusion equation with potential ut= um V (x)um+upin Rn×(0,T), where 1 2αm+n< m ≤ 1, p > 1, n ≥ 2, V (x) ～ω|x|2with ω≥ 0 as |x| →∞,and α is the positive root of αm(αm + n 2) ω = 0. The critical Fujita exponent was determined as pc= m +2αm+nin a previous paper of the authors. In the present paper,we establish the second critical exponent to identify the global and non-global solutions in their co-existence parameter region p > pcvia the critical decay rates of the initial data.With u0(x) ～ |x| aas |x| →∞, it is shown that the second critical exponent a =2p m,independent of the potential parameter ω, is quite different from the situation for the critical exponent pc.     

Stability of planar diffusion wave for nonlinear evolution equation

It is known that the one-dimensional nonlinear heat equation ut = f(u)x1x1,f’(u) > 0,u(±∞,t) = u±,u+ = u_ has a unique self-similar solution u(x1/1+t).In multi-dimensional space,u(x1/1+t) is called a planar diffusion wave.In the first part of the present paper,it is shown that under some smallness conditions,such a planar diffusion wave is nonlinearly stable for the nonlinear heat equation:ut-△f(u) = 0,x ∈ Rn.The optimal time decay rate is obtained.In the second part of this paper,it is further shown that this planar diffusion wave is still nonlinearly stable for the quasilinear wave equation with damping:utt + utt+ △f(u) = 0,x ∈ Rn.The time decay rate is also obtained.The proofs are given by an elementary energy method.     

具有非线性记忆项的耗散波动方程的整体吸引子

LetΩRn be a bounded domain with a smooth boundary.We consider the longtime dynamics of a class of damped wave equations with a nonlinear memory term utt+αut-△u-∫0t 0μ(t-s)|u(s)| βu(s)ds + g(u)=f.Based on a time-uniform priori estimate method,the existence of the compact global attractor is proved for this model in the phase space H10(Ω)×L2(Ω).     

ON THE CAUCHY PROBLEM FOR AW-RASCLE SYSTEM WITH LINEAR DAMPING

The existence of global BV solutions for the Aw-Rascle system with linear damping is considered.In order to get approximate solutions we consider the system in Lagrangian coordinates,then by using the wave front tracking method coupling with and suitable splitting algorithm and the ideas of[1]we get a sequence of approximate solutions.Finally we show the convergence of this approximate sequence to the weak entropic solution.     

KAM Tori for the Derivative Quintic Nonlinear Schrodinger Equation

This paper is concerned with one-dimensional derivative quintic nonlinear Schrodinger equation,iut—uxx+i(|u|4u)x=0,x eT.The existence of a large amount of quasi-periodic solutions with two frequencies for this equation is established.The proof is based on partial Birkhoff normal form technique and an unbounded KAM theorem.We mention that in the present paper the mean value of u does not need to be zero,but small enough,which is different from the assumption(1.7)in Geng-Wu[J.Math.Phys.、53,102702(2012)].     

Solitary Wave Solutions of Delayed Coupled Higgs Field Equation

This paper is devoted to the study of the solitary wave solutions for the delayed coupled Higgs field equation{vtt-uxx-αu+βf*u|u|²-2uv-τu(|u|²)x=0 vtt+vxx-β(|u|^x)xx=0.We first establish the existence of solitary wave solutions for the corresponding equation without delay and perturbation by using the Hamiltonian system method.Then we consider the persistence of solitary wave solutions of the delayed coupled Higgs field equation by using the method of dynamical system,especially the geometric singular perturbation theory,invariant manifold theory and Fredholm theory.According to the relationship between solitary wave and homoclinic orbit,the coupled Higgs field equation is transformed into the ordinary differential equations with fast variables by using the variable substitution.It is proved that the equations with perturbation also possess homoclinic orbit,and thus we obtain the existence of solitary wave solutions of the delayed coupled Higgs field equation.     

二阶常微初值问题的单步格式

讨论二阶常微分方程初值问题utt+au=f,u(0)=u0，ut(0)=v0 的一种单步格式，采用u及v=ut为未知量，计算简单. 证明了此格式的稳定性及对u，v皆有二阶精度. 此格式可用于双曲问题.     

Entropy Unilateral Solution for Some Noncoercive Nonlinear Parabolic Problems Via a Sequence of Penalized Equations

We give an existence result of the obstacle parabolic equations(b(x,u))/(t)-div(a(x,t,u,▽u))+div(φ(x,t,u))=f in Q_T,where b(x,u) is bounded function of u,the term-div(a(x,t,u,▽u)) is a Leray-Lions type operator and the function φ is a nonlinear lower order and satisfy only the growth condition.The second term f belongs to L~1(Q_T).The proof of an existence solution is based on the penalization methods.     

超线性非线性Sturm-Liouville边值问题的正解

利用拓扑方法讨论了一类非线性Sturm-Liouville边值问题{-u″=λf(x,u),0≤x≤1,α0u(0)+β0u′(0)=0, α1u(1)+β1u′(1)=0.研究了上述问题的正解的全局结构,在非线性项f(x,u)不满足条件f(x,u)≥0(u≥0)时获得了正解的存在性.     

Multiple Sign-Changing Solutions for Quasilinear Equations of Bounded Quasilinearity

The existence of an infinite sequence of sign-changing solutions are proved for a class of quasilinear elliptic equations under suitable conditions on the quasilinear coefficients and the nonlinearity■ where ? ? R~N is a bounded domain with smooth boundary, and we use■ The main interest of this paper is for the case of bounded quasilinearity bij. The result is proved by an elliptic regularization method involving truncations of both u and the gradient of u.     

一类三维微分系统正值解的结构(英文)

研究三维微分系统:u′1=a1(t)|u2|λ1sgn u2,u′2=a2(t)|u3|λ2sgn u3,u′3=-a3(t)|u1|λ3sgn u1.假设λi(i=1,2,3)是正的常数,ai(t)(i=1,2,3)在区间[0,∞)上是正的连续函数,根据u的分量ui的特殊渐近条件定义了正值解的几种类型。系统满足条件∫0∞ai(t)dt=∞,i=1,2.     

具p-Laplacian算子的四阶三点边值问题的迭代解的存在性

研究下列具有p-Laplacian算子的四阶三点边值问题{(φp(u″(t)))″=f(t,u(t),u″(t)),t∈[0,1] u(0)-ξu(1)=0,u′(1)-ηu′(0)=0 u″(0)-a1u″(δ)=0,(φp(u″))′(1)-b1(φp(u″))′(δ)=0其中φp(s)=|s|p-2s,p>1,0<ξ,η<1,0相似文献   

关于一类非线性微分方程组的边值问题的渐近解（Ⅰ）

应用新的方法 ,研究一类非线性微分方程组　　u″ =v,εv″ f(x ,u ,u′)v′ -g(x ,u ,u′)v =0　　 (0 <ε 1) ,的边值问题的解的渐近性质·　构造出解的渐近展开式 ,和估计了余项 ,改进并拓广了前人的工作·     

图的Hyper-Wiener指数的综述(英文)

设G是一个图.G的顶点u和v的距离是u和v之间最短路的长度.Wiener指数是G中所有无序顶点对之间距离之和,而Hyper-Wiener指数定义为WW(G)=?∑u,v∈V(G)d(u,v)+?∑u,v∈V(G)d2(u,v),式中的和取遍G的所有顶点对.本文总结了图的Hyper-Wiener指数的最近结论.     

缺乏单调性的奇异三阶三点边值问题的正解(英文)

考虑了非线性三阶三点边值问题u′′′(t)=f(t,u(t))+g(t,u(t)),0t1,u(0)=u′(η)=u′′(1)=0的正解.本文中g(t,u)可以在t=0,t=1及u=0处奇异,而且允许g(t,u)关于u不是非减的.通过构造适当的控制函数并且利用锥上的Guo-Krasnoselskii不动点定理,证明了若干新的局部存在定理.     

U-标算子的连续演算

本文给出了U-标算子经连续算子演算后有界的充分条件。另外,对一给定的U-标算子T,证明了当函数μ(t)连续可导时算子μ(T)有界,但存在连续函数使得μ(T)无界。