The Almost Global and Global Existence for Quasi-linear Wave Equations with Multiple-Propagation Speeds in High Dimensions

In this paper, we consider the Cauchy problem for systems of quasi-linear wave equations with multiple propagation speeds in spatial dimensions n ≥ 4. The problem when the nonlinearities depend on both the unknown function and their derivatives is studied. Based on some Klainerman- Sideris type weighted estimates and space-time L2 estimates, the results that the almost global existence for space dimensions n = 4 and global existence for n≥ 5 of small amplitude solutions are presented.

Wave equations and reaction-diffusion equations with several nonlinear source terms

The initial boundary value problem of wave equations and reaction-diffusion equations with several nonlinear source terms in a bounded domain is studied by potential well method. The invarianee of some sets under the flow of these problems and the vacuum isolation of solutions are obtained by introducing a family of potential wells. Then the threshold result of global existence and nonexistence of solutions are given. Finally, the problem with critical initial conditions are discussed.

A Global Existence Theorem for Semilinear Wave Equations in Five Space Dimension

This paper concerns the global existence of solutions to the semi-linear wave equation utt-△u = G(u) in five space dimensions, where G(u) -|u|p with p > 3+17^(1/17). We used the classical iteration method and technique estimates to show that a classical global solution exists for the radially symmetric equations with small and compact supported initial data.

THE EXISTENCE OF GLOBAL SOLUTION AND "BLOW UP" PHENOMENON FOR A SYSTEM OF MULTI DIMENSIONAL SYMMETRIC REGULARIZED WAVE EQUATIONS

The existence and uniqueness of the global smooth solution to the initial-boundary valueproblem of a system of multi-dimensions SRWE are proved. The sufficient conditions of 'blowingup' of the solution are given.

GLOBAL EXISTENCE FOR A CLASS OF SYSTEMSOF NONLINEAR WAVE EQUATIONS INTHREE SPACE DIMENSIONS

Consider a system of nonlinear wave equations (e)2t-c2i△xui=Fi(u,(e)u,(e)x(e)u) in (0,∞)×(R)3 for i=1,┈,m,where Fi(i=1,┈,m) are smooth functions of degree 2 near the origin of their arguments, and u=(u1,┈,um),while (e)u and (e)x(e)u represent the first and (second derivatives of u, respectively. In this paper, the author presents a new class of nonlinearity for which the global existence of small solutions is ensured. For example, global existence of small solutions for((e)2t- c21Δx)u1 = u2((e)tu2) + arbitrary cubic terms,((e)2t - c22Δx)u2=u1((e)tu2) + ((e)tu1)u2 + arbitrary cubic termswill be established, provided that c21 ≠ c22.

SIMILARITY TRANSFORMATION, THE STRUCTURE OF THE TRAVELING WAVES SOLUTION AND THE EXISTENCE OF A GLOBAL SMOOTH SOLUTION TO GENERALIZED KURAMOTO SIVASHINSKY TYPE EQUATIONS

The existence of a global smooth solution for the initial value problem of generalized Kuramoto-Sivashinsky type equations have been obtained. Similarty siolutions and the structure of the traveling waves solution for the generalized KS equations are discussed and analysed by using the qualitative theory of ODE and Lie's infinitesimal transformation respectively.

GLOBAL EXISTENCE AND EXPONENTIAL STABILITY OF SOLUTIONS TO THE QUASILINEAR THERMO-DIFFUSION EQUATIONS WITH SECOND SOUND

This article is devoted to the study of global existence and exponential stability of solutions to an initial-boundary value problem of the quasilinear thermo-diffusion equations with second sound by means of multiplicative techniques and energy method provided that the initial data are close to the equilibrium and the relaxation kernel is strongly positive definite and decays exponentially.

GLOBAL EXISTENCE AND L^p ESTIMATES FOR SOLUTIONS OF DAMPED WAVE EQUATION WITH NONLINEAR CONVECTION IN MULTI-DIMENSIONAL SPACE

In this article, the author studies the Cauchy problem of the damped wave equation with a nonlinear convection term in multi-dimensions. The author shows that a classical solution to the Cauchy problem exists globally in time under smallness condition on the initial perturbation. Furthermore, the author obtains the L^p （2 ≤ p ≤ ∞） decay estimates of the solution.

Semi-linear Wave Equations with Effective Damping

The authors study the Cauchy problem for the semi-linear damped wave equation utt-△u＋b（t）ut=f（u）,u（0,χ）=u0（χ）,ut（0,χ）=u1（χ） in any space dimension n ≥ 1. It is assumed that the time-dependent damping term b（t）〉 0 is effective, and in particular tb（t） →∞ as t →∞. The global existence of small energy data solutions for｜f（u）｜≈｜u｜^p in the supercritical case of p 〉 1＋ 2/n and p ≤n/n-2 for n ≥ 3 is proved.

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.

Global Existence and Blow-up for Semilinear Wave Equations with Variable Coefficients

The authors consider the critical exponent problem for the variable coefficients wave equation with a space dependent potential and source term. For sufficiently small data with compact support, if the power of nonlinearity is larger than the expected exponent, it is proved that there exists a global solution. Furthermore, the precise decay estimates for the energy, L^2 and L^(p+1) norms of solutions are also established. In addition, the blow-up of the solutions is proved for arbitrary initial data with compact support when the power of nonlinearity is less than some constant.

具色散非线性波动方程解的存在性与爆破

研究了一类非线性波动方程初边值问题整体解的存在性与爆破问题.利用位势井方法证明了具有两个异号源和具有四阶色散项的波动方程在方程具有负定能量情形下初边值问题整体解的存在性.利用凹性方法证明了具有任意初始正能量时解的有限时间爆破问题.

THE CAUCHY PROBLEM FOR THE CAMASSA-HOLM-NOVIKOV EQUATION

In this paper,we consider the Cauchy problem for the Camassa-Holm-Novikov equation.First,we establish the local well-posedness and the blow-up scenario.Second,infinite propagation speed is obtained as the nontrivial solution u(x,t)does not have compact x-support for any t>0 in its lifespan,although the corresponding u0(x)is compactly supported.Then,the global existence and large time behavior for the support of the momentum density are considered.Finally,we study the persistence property of the solution in weighted Sobolev spaces.

具有多个非线性源项的波动方程

利用位势井方法研究在有界区域上具有多个非线性源项的波动方程初边值问题.给出了位势井的结构和位势井深度函数的性质.通过引进位势井族得到了在这些问题的流之下的一些集合不变性以及解的真空隔离,揭示了只要问题的初值属于位势井内或位势井外,则问题在今后所有时间内的解都存在于位势井内或井外,同时存在一个没有解的空间区域.进而给出了解的整体存在和不存在的门槛结果.最后,利用相同的方法讨论了具有临界初始条件的问题.

三维空间中一类非线性波动方程整体解的存在性

研究了非线性波动方程utt+but =Δu+F(u) ,t >0 ,x∈R3 ,u(0 ,x) =f(x) ,ut(0 ,x) =g(x) ,x∈R3 的整体解 ,其中Δ =∑3i=1 2 x2 i,常数b >0 ,得到了波动方程整体解的存在性 .

一类非线性色散耗散波动方程整体解的存在性

研究一类具有色散项与耗散项的四阶非线性波动方程在n维空间中有界域上的Dirichlet初边值问题.其中半线性项f(u)与u的符号相同,并满足一定的增长条件.首先定义了位势井W与井外集合V,并给出了它们的一些基本性质,而后证明了若初始能量小于位势井深度d且初始函数在位势井W中,此问题存在一个整体弱解,且这个解也在位势井W中,最后证明了若初始函数在空间的某个球内,此问题也存在一个整体弱解,且这个解也在该球内.

位势井及其对具异号源项波动方程的应用

本文研究具有两个异号非线性源项的波动方程的初边值问题。应用位势井方法,解决了不具正定能量情况下问题整体解的存在性问题。证明了对于非线性项的指数在一定条件下,问题存在整体弱解。从而拓展了已知结果。

一类非线性Schrdinger方程组的整体解和爆破解

考虑了一类非线性Schr dinger方程组的柯西问题iβt+mΔ=c(p+1)||p-1|ψ|q+1,t>0,x∈R2itψ+sΔψ=b(q+1)|ψ|q-1||p+1ψ,t>0,x∈R2,根据基态的驻波的存在和局部理论,用势井方法和凹函数方法给出了它的爆破解和整体解存在的最佳条件.

四阶具强阻尼非线性波动方程解的整体存在性与不存在性

研究四阶具强阻尼非线性波动方程utt-△u＋△^2u-α△ut=f（u）的初边值问题.利用位势井方法,得到了问题整体解存在性与不存在性的最佳条件（门槛结果）.

一类非线性Kirehhoff波方程的整体解

研究了一类非线性Kirchhoff波方程utt-M(‖u‖22)△u+g(ut)(t)+f(u)=0的初边值问题.在一定条件下,作者证明了方程整体解的存在唯一性,得到了解的衰减估计.