A NOTE ON LARGE TIME BEHAVIOUR OF SOLUTIONS FOR VISCOUS CAHN-HILLIARD EQUATION

This article is devoted to the discussion of large time behaviour of solutions for viscous Cahn-Hilliard equation with spatial dimension n 〈 5. Some results on global existence of weak solutions for small initial value and blow-up of solutions for any nontrivial initial value are established.     

The pointwise estimates of solutions for a nonlinear convection diffusion reaction equation

This paper studies the time asymptotic behavior of solutions for a nonlinear convection diffusion reaction equation in one dimension.First,the pointwise estimates of solutions are obtained,furthermore,we obtain the optimal L~p,1≤ p ≤ +∞,convergence rate of solutions for small initial data.Then we establish the local existence of solutions,the blow up criterion and the sufficient condition to ensure the nonnegativity of solutions for large initial data.Our approach is based on the detailed analysis of the Green function of the linearized equation and some energy estimates.     

具有边界摄动的非线性非局部反应扩散方程的奇摄动问题

A class of nonlinear nonlocal singularly perturbed Robin initial boundary value problems for reaction diffusion equations with boundary perturbation is considered. Under suitable conditions, firstly, the outer solution of the original problem is obtained; secondly, by using the stretched variable, the composing expansion method and the expanding theory of power series, the initial layer is constructed; and finally, by using the theory of differential inequalities the asymptotic behavior of solutions for initial boundary value problems is studied, and including some relational inequalities the existence and uniqueness of solutions for the original problem and the uniformly valid asymptotic estimation are discussed.     

The Cauchy Problems and Global Solutions to the Deybe System and Burgers＇ Equations in Pseudomeasure Spaces

In this paper we consider the Cauchy problems of Burgers＇ equations and the Deybe system. Their existence and uniqueness of the time-global solutions for small initial data in some pseudomeasure spaces are obtained. The asymptotic stability of small solutions is proved. As an immediate result the existence and uniqueness of the self-similar solutions are also obtained provided the initial data satisfy the self-similar structures.     

FORMATION OF SINGULARITY FOR COMPRESSIBLE VISCOELASTICITY Dedicated to Professor Constantine M. Dafermos on the occasion of his 70th birthday

The formation of singularity and breakdown of classical solutions to the threedimensional compressible viscoelasticity and inviscid elasticity are considered.For the compressible inviscid elastic fluids,the finite-time formation of singularity in classical solutions is proved for certain initial data.For the compressible viscoelastic fluids,a criterion in term of the temporal integral of the velocity gradient is obtained for the breakdown of smooth solutions.     

Global positive periodic solutions of age-dependent competing systems

This paper is to investigate positive periodic solutions of a biological system composed of two competing species. The existence and uniqueness of nonnegative solutions to the model for a set of given vital rates and initial distribution are treated and the contractive property of the solutions explored. Based on these results, some simple conditions for the global existence of positive periodic orbits are established by means of Horn’s asymptotic fixed point theorem.     

A REMARK ON LARGE TIME ASYMTOTICS FOR SOLUTIONS OF A NONHOMOGENEOUS VISCOUS BURGERS EQUATION

The existence, uniqueness and regularity of solutions to the Cauchy problem posed for a nonhomogeneous viscous Burger’s equation were shown in Chung, Kim and Slemrod [1] by assuming suitable conditions on initial data. Moreover, they derived the asymptotic behaviour of solutions of the Cauchy problem by imposing additional conditions on initial data. In this article, we obtain the same asymptotic behaviour of solutions to the Cauchy problem without imposing additional condition on initial data.     

Monotone method for initial value problem for fractional diffusion equation

Using the method of upper and lower solutions and its associated monotone iterative, consider the existence and uniqueness of solution of an initial value problem for the nonlinear fractional diffusion equation.     

Equivalence Relation between Initial Values and Solutions for Evolution p-Laplacian Equation in Unbounded Space

In this paper,an equivalence relation between the ω-limit set of initial values and the ω-limit set of solutions is established for the Cauchy problem of evolution p-Laplacian equation in the unbounded space Yσ(RN).To overcome the difficulties caused by the nonlinearity of the equation and the unbounded solutions,we establish the propagation estimate and the growth estimate for the solutions.It will be demonstrated that the equivalence relation can be used to study the asymptotic behavior of solutions.     

Upper bounds of the rates of decay for solutions of the Boussinesq equations

In this paper,upper bounds of the L2-decay rate for the Boussinesq equations are considered.Using the L2 decay rate of solutions for the heat equation,and assuming that the solutions of the Boussinesq equations are smooth,we obtain the upper bounds of L2 decay rate for the smooth solutions and difference between the solutions of the Boussinesq equations and those of the heat system with the same initial data.The decay results may then be obtained by passing to the limit of approximating sequences of solutions.The main tool is the Fourier splitting method.     

STRONG SOLUTIONS FOR THE INCOMPRESSIBLE FLUID MODELS OF KORTEWEG TYPE

In this article, we are concerned with the strong solutions for the incompressible fluid models of Korteweg type in a bounded domain Ω R3. We prove the existence and uniqueness of local strong solutions to the initial boundary value problem. We point out that in this article we allow the existence of initial vacuum provided initial data satisfy a compatibility condition.     

DISCONTINUOUS SOLUTIONS IN L^∞ FOR HAMILTON-JACOBI EQUATIONS

An approach is introduced to construct global discontinuous solutions in L∞ for Hamilton-Jacobi equations. This approach allows the initial data only in L∞ and applies to the equations with nonconvex Hamiltonians. The profit functions are introduced to formulate the notion of discontinuous solutions in L∞. The existence of global discontinuous solutions in L∞ is established. These solutions in L∞ coincide with the viscosity solutions and the minimax solutions, provided that the initial data are continuous. A prototypical equation is analyzed toexamine the L∞ stability of our L∞ solutions. The analysis also shows that global discontinuous solutions are determined by the topology in which the initial data are approximated.     

Convergence of Compressible Euler-Maxwell Equations to Compressible Euler-Poisson Equations

In this paper,the convergence of time-dependent Euler-Maxwell equations to compressible Euler-Poisson equations in a torus via the non-relativistic limit is studied. The local existence of smooth solutions to both systems is proved by using energy esti- mates for first order symmetrizable hyperbolic systems.For well prepared initial data the convergence of solutions is rigorously justified by an analysis of asymptotic expansions up to any order.The authors perform also an initial layer analysis for general initial data and prove the convergence of asymptotic expansions up to first order.     

Lp Decay Rate for a Nonlinear Convection Diffusion Reaction Equation in R~n

This paper studies the asymptotic behavior of solutions for a nonlinear convection diffusion reaction equation in Rⁿ.Firstly,the global existence and uniqueness of classical solutions for small initial data are established.Then,we obtain the L^p,2≤p≤+∞decay rate of solutions.The approach is based on detailed analysis of the Green function of the linearized equation with the technique of long wave-short wave decomposition and the Fourier analysis.     

A NOTE ON THE INTERACTIONS OF ELEMENTARY WAVES FOR THE AR TRAFFIC FLOW MODEL WITHOUT VACUUM

In this note, we consider the interactions of elementary waves for the traffic flow model proposed by Aw and Rascle when the vacuum is not involved. The solutions are obtained constructively and globally when the initial data consist of three pieces of constant states. Furthermore, it can be found that the Riemann solutions are stable with respect to such small perturbations of the initial data in this particular situation by investigating the limits of the solutions as the perturbed parameter ε goes to zero.     

Existence and Nonuniqueness of Weak Solutions of the Initial-Boundary Value Problem for u_t = u~σdiv(|u|~(p-2)u)

In this paper, we study the existence and nonuniqueness of weak solutions of the initial and boundary value problem for ut= uσdiv(|△u|p-2△u) with σ≥1. Localization property of weak solutions is also discussed.     

INITIAL BOUNDARY VALUE PROBLEMS FOR SEMILINEAR WAVE EQUATIONS WITH DISCONTINUOUS DATA

This paper studies the initial boundary problems for semilinear strictly hyperbolic second-order equations with discontinuous data. Under the uniform Lopatinski boundary condition, the local existence theorem of discontinuous solutions and the structure of singularities of the solutions are obtained.     

Asymptotic of the Solutions of the Initial Boundary Value Problem for the Diffusion Equations for Semiconductors（Ⅱ）

The paper deal with the asymptotic behavior of the solutions to the initial boundary value problem for unipolar drift diffusion equations for semiconductors. Under the proper assumptions on doping profile and initial value, we prove that the smooth solutions to these evolutionary problems tend to the unique stationary solution exponentially as time tends to infinity.     

UNIQUENESS OF VISCOSITY SOLUTIONS OF FULLY NONLINEAR SECOND ORDER PARABOLIC PDE''''S

This paper investigates the maximum principle for viscosity solutions of fully nonlinear, second order parabolic, possibly degenerate partial differential equations. Using this maximum principle the anthor prove that viscosity solutions of initial and bo unoary value problem for parabolic equations are unique.     

EXISTENCE AND DECAY OF WEAK SOLUTIONS TO DEGENERATE DAVEY-STEWARTSON EQUATIONS

In this paper, the authors consider a system of degenerate Davey-Stewartson equations. They prove the global existence of weak solutions in some weighted function spaces and the decay of weak solutions in some anisotropic spaces for appropriate initial data.