Asymptotic Properties for a Semilinear Edge-Degenerate Parabolic Equation

The author deals with a semi-linear edge-degenerate parabolic equation, and proves that the solution increases exponentially under the initial energy J(u_0) ≤ d, where d is the mountain-pass level. Moreover, the author estimates the blow-up time and the blow-up rate for the solution under J(u_0) 0.     

一类非线性波方程整体解的存在性和不存在性

This paper deals with the existence and uniqueness of the global solution of the initial boundary value problem of a class of wave equation. In the meantime, it gives the sufficient conditions of blow-up of the solution for the problem in finite time.     

一类具强阻尼Kirchhoff方程的能量衰减和解的爆破

The initial boundary value problem for a Kirchhoff equation with Lipschitz type continuous coefficient is studied on bounded domain. Under some conditions, the energy decaying and blow-up of solution are discussed. By refining method, the exponent decay estimates of the energy function and the estimates of the life span of blow-up solutions are given.     

Notes on Global Existence for the Nonlinear Schrdinger Equation Involves Derivative

In this paper, we consider the scattering for the nonlinear Schr¨odinger equation with small,smooth, and localized data. In particular, we prove that the solution of the quadratic nonlinear Schr¨odinger equation with nonlinear term |u|2involving some derivatives in two dimension exists globally and scatters. It is worth to note that there exist blow-up solutions of these equations without derivatives. Moreover, for radial data, we prove that for the equation with p-order nonlinearity with derivatives, the similar results hold for p ≥2d+32d-1and d ≥ 2, which is lower than the Strauss exponents.     

GLOBAL EXISTENCE AND EXPONENTIAL DECAY OF SOLUTIONS OF GENERALIZED KURAMOTO-SIVASHINSKY EQUATIONS

We study the Dirichlet initial-boundary value problem of the generalized Kuramoto-Sivashinsky equation ut+uxxxx+λuxx+f(u)x=0 on the interval [0,l],The nonlinear function f satisfies the conditon |f′(u)|≤c|u|^α-1 for some α>1. We prove that if λ4π^2/t^2,then the strong solution is global and exponentially decays to zero for and initial datum uo∈H0^2(0,l) if 1<α≤7,and for small u0∈H0^2(0,l)if α>7,We the consider the equation ut+uxxxx+λuzz+μu+auxxx+bux=F(u,ux,uxx,uxxx),We prove that if F is twice differentiable,Δ↓F is Lipschitz continuous,and F(0)=Δ↓F（0)=0,and if λand μsatisfu μ+σ(λ）>0(σ（λ）=the first eigenvalue of the operator d^4/dx^4+λd^2/dx^2),then the solution for small initial datum is global and exponentially decays to zero.     

Scattering Versus Blowup Beyond the Mass-Energy Threshold for the Davey–Stewartson Equation in R~3

We study the Cauchy problem for the Davey–Stewartson equation i?_tu + Δu + |u|~2 u + E_1(|u|~2)u = 0,(t, x) ∈ R × R~3.The dichotomy between scattering and finite time blow-up shall be proved for initial data with finite variance and with mass-energy M(u_0)E(u_0) above the ground state threshold M(Q)E(Q).     

EXISTENCE OF INFINITE ENERGY SOLUTION TO THE INELASTIC BOLTZMANN EQUATION WITH EXTERNAL FORCE

In this paper,the Cauchy problem for the inelastic Boltzmann equation with external force is considered in the case of initial data with infinite energy.More precisely,under the assumptions on the bicharacteristic generated by external force,we prove the global existence of solution for small initial data compared to the local Maxwellian exp{ p|x v|2},which has infinite mass and energy.     

BLOW-UP RESULTS AND ASYMPTOTIC BEHAVIOR OF THE EMDEN-FOWLER EQUATION u″=|u|~p

In this article the author works with the ordinary differential equation u″= |u|~p for some p>0 and obtains some interesting phenomena concerning blow-up,blow-up rate,life-span,stability,instability,zeros and critical points of solutions to this equation.     

一类退化抛物方程高初始能量下解的有限时刻爆破及整体存在性（英文）

We consider the initial-boundary value problem for finitely degenerate parabolic equation. We first give sufficient conditions for the blow-up and global existence of the parabolic equation at high initial energy level. Then, we establish the existence of solutions blowing up in finite time with initial data at arbitrary energy level. Finally, we estimate the upper bound of the blow-up time under certain conditions.     

Local and Global Existence of Solutions to Initial Value Problems of Nonlinear Kaup-Kupershmidt Equations

This paper is devoted to studying the initial value problems of the nonlinear Kaup Kupershmidt equations δu/δt ＋ α1 uδ^2u/δx^2 ＋ βδ^3u/δx^3 ＋ γδ^5u/δx^5 = 0, （x,t）∈ E R^2, and δu/δt ＋ α2 δu/δx δ^2u/δx^2 ＋ βδ^3u/δx^3 ＋ γδ^5u/δx^5 = 0, （x, t） ∈R^2. Several important Strichartz type estimates for the fundamental solution of the corresponding linear problem are established. Then we apply such estimates to prove the local and global existence of solutions for the initial value problems of the nonlinear Kaup- Kupershmidt equations. The results show that a local solution exists if the initial function u0（x） ∈ H^s（R）, and s ≥ 5/4 for the first equation and s≥301/108 for the second equation.     

Periodic Solutions for a Class of Forced Liénard-type Equations

Abstract   By applying the topological degree theory, we establish some sufficient conditions for the existence on T-periodic solutions for the Liénard-type equation

Precise Asymptotics in the Law of the Iterated Logarithm of Moving-Average Processes

In this paper, we discuss the moving-average process Xk = ∑i=-∞ ^∞ ai＋kεi, where {εi;-∞ 〈 i 〈 ∞} is a doubly infinite sequence of identically distributed ψ-mixing or negatively associated random variables with mean zeros and finite variances, {ai;-∞ 〈 i 〈 -∞） is an absolutely solutely summable sequence of real numbers.     

A Littlewood-Paley type inequality

In this note we prove the following theorem: Let u be a harmonic function in the unit ball and . Then there is a constant C = C(p, n) such that

Cauchy problem for a semilinear Éidel’man parabolic equation

We establish conditions for the existence and uniqueness of a generalized solution of the Cauchy problem for the equation

Equivalent definition of some weighted Hardy spaces

We present an equivalent definition of functions analytic in the half-plane ℂ₊ = {z: Re z > 0} for which

Estimate of fourier transforms with respect to the system of generalized eigenfunctions of the Schrödinger operator with Stummel-type potential

Suppose thatА is a nonnegative self-adjoint extension to { } of the formal differential operator−Δu+q(x)u with potentialq(x) satisfying the condition {

The Commutators of Fractional Integrals on Besov Spaces

Abstract   In this paper we consider the commutators of fractional integrals

On solutions of almost hypoelliptic equations in weighted Sobolev spaces

It is proved that if P(D) is a regular, almost hypoelliptic operator and

On the riemann zeta-function and the divisor problem II

Let Δ(x) denote the error term in the Dirichlet divisor problem, and E(T) the error term in the asymptotic formula for the mean square of . If E ^*(t)=E(t)-2πΔ^*(t/2π) with , then we obtain

A note on regularity for a class of quasilinear elliptic equations

The following regularity of weak solutions of a class of elliptic equations of the form are investigated.