关于一类具阻尼项的非线性波方程的Cauchy问题

The paper concerns with the existence, uniqueness and nonexistence of global solution to the Cauchy problem for a class of nonlinear wave equations with damping term. It proves that under suitable assumptions on nonlinear the function and initial data the above-mentioned problem admits a unique global solution by Fourier transform method. The sufficient conditions of nonexistence of the global solution to the above-mentioned problem are given by the concavity method.     

一类广义长短波方程的整体适定性

In the present paper,we investigate the well-posedness of the global solution for the Cauchy problem of generalized long-short wave equations.Applying Kato's method for abstract quasi-linear evolution equations and a priori estimates of solution,we get the existence of globally smooth solution.     

GLOBAL EXISTENCE OF SOLUTIONS OF CAUCHY PROBLEM FOR GENERALIZED BBM-BURGERS EQUATION

In this paper, the existence and the uniqueness of the local generalized solution and the local classical solution of the Cauchy problem for the generalized BBM-Burgers equationare proved. The existence and the uniqueness of the global generalized solution and the global classical solution for the Cauchy problem of equation （1） are proved when n = 3, 2, 1. Moreover, the decay property of the solution is discussed.     

Dynamical System Method for Solving Ill-Posed Operator Equations

Two dynamical system methods are studied for solving linear ill-posed problems with both operator and right-hand nonexact. The methods solve a Cauchy problem for a linear operator equation which possesses a global solution. The limit of the global solution at infinity solves the original linear equation. Moreover, we also present a convergent iterative process for solving the Cauchy problem.     

Global regularity for 3D magneto-hydrodynamics equations with only horizontal dissipation

We investigate the Cauchy problem for the 3D magneto-hydrodynamics equations with only horizontal dissipation for the small initial data. With the help of the dissipation in the horizontal direction and the structure of the system, we analyze the properties of the decay of the solution and apply these decay properties to get the global regularity of the solution. In the process, we mainly use the frequency decomposition in Green's function method and energy method.     

PERIODIC BOUNDARY VALUE PROBLEM AND CAUCHY PROBLEM OF THE GENERALIZED CUBIC DOUBLE DISPERSION EQUATION

In this article, the existence, uniqueness and regularities of the global generalized solution and global classical solution for the periodic boundary value problem and the Cauchy problem of the general cubic double dispersion equation
utt - uxx - auxxtt ＋ bux4 - duxxt = f（u）xx
are proved, and the sufficient conditions of blow-up of the solutions for the Cauchy problems in finite time are given.     

Existence and uniqueness of global classical solutions to 3D isentropic compressible Navier-Stokes equations with general initial data

In this paper we study the global existence and uniqueness of classical solutions to the Cauchy problem for 3D isentropic compressible Navier-Stokes equations with general initial data which could contain vacuum.We give the relation between the viscosity coefficients and the initial energy,which implies that the Cauchy problem under consideration has a global classical solution.     

EXISTENCE OF GLOBAL SMOOTH SOLUTION TO JIN-XIN MODEL WITH LARGE INITIAL DATA

In this paper, Under the assumption that the relaxation time ε is sufficiently small, we prove the existence of the global smooth solution to the Cauchy problem for the Jin-Xin model without any smallness assumption for the initial data.The analysis is based on some a priori estimates which are obtained by the method of characteristic and the maximum principle of first-order quasilinear hyperbolic system.     

CAUCHY PROBLEM FOR A NONLINEAR SINGULAR INTEGRAL-DIFFERENTIAL EVOLUTION SYSTEM

We study the Cauchy problem for a nonlinear evolution system with singularintegral differential terms, By means of some a priori estimates of the solution and the Leray-Schander‘s fixed point theorem, we prove the existence and the uniqueness theorems of the generalized global solution of the mentioned problem.     

ASYMPTOTIC STABILITY OF RAREFACTION WAVE FOR HYPERBOLIC-ELLIPTIC COUPLED SYSTEM IN RADIATING GAS

In this article, authors study the Cauch problem for a model of hyperbolic-elliptic coupled system derived from the one-dimensional system of the radiating gas. By considering the initial data as a small disturbances of rarefaction wave of inviscid Burgers equation, the global existence of the solution to the corresponding Cauchy problem and asymptotic stability of rarefaction wave is proved. The analysis is based on a priori estimates and L2-energy method.     

Cauchy problem of the generalized double dispersion equation

In this paper, the existence and the uniqueness of the global solution for the Cauchy problem of the generalized double dispersion equation are proved. The blow-up of the solution for the Cauchy problem of the generalized double dispersion equation is discussed by the concavity method under some conditions.     

The Cauchy Problem for the Generalized IMBq Equation in W(R)

In this paper, the existence and the uniqueness of the global strong solution and the global classical solution for the Cauchy problem of the multidimensional generalized IMBq equation are proved. The nonexistence of the global solution for the Cauchy problem of the generalized IMBq equation is discussed.     

Global Solutions and Their Large-time Behavior of Cauchy Problem for Equations of Deep Water Type

We consider the large time behavior of the global solution of the Cauchy problem for the equation of Benjamin-Ono type. A series of large-time global estimates for Cauchy problem arc constructed. By means of the obtained global estimates uniformly for 0 ≤ t < ∞, the attractors of the Cauchy problem for the mentioned nonlinear equations are considered. And also the dimensions of the global attractor are estimated.     

Cauchy Problem for a Generalized Nonlinear Dispersive Equation

The solvability of global smooth solution for the Cauchy problem of a generalized nonlinear dispersive equation is studied by using the continuation method. In addition, the convergences of solution for this problem are also discussed.     

解算子与右端数据均有扰动的半正定算子方程的动态系统方法

本文研究了求解算子与右端数据均有扰动的第一类半正定算子方程的动态系统方法．证明了相应的动态系统Cauchy问题的整体解存在且收敛于原算子方程的解．此外,给出了解Cauchy问题的迭代方法并证明了方法的收敛性．     

一类具阻尼非线性波动方程的Cauchy问题

本文研究一类具阻尼非线性波动方程Cauchy问题整体广义解和整体古典解的存在唯一性,并用凸性方法给出解爆破的充分条件.