Abstract : In this paper we will give a classification of radial solutions of some biharmonic equations.

Abstract : Impulsive partial differential systems can be successfully used for mathematical simulation in theory physics, Chemistry, biotechnology, medicine, population dynamics, optimal control, and in other process and phenomena and technology. There has been increasing interest in impulsive partial differential systems during the past few years, and several papers concerning the qualitative theory of impulsive partial differential systems without delay have appeared recently. However, very little is known about impulsive partial differential systems with delay, and so far there are no results concerning the oscillation theory of impulsive partial differential systems with delay, as far as we know. The objective of this talk to investigate the oscillation properties of the solutions of a class of nonlinear impulsive parabolic systems with delay. We establish several oscillation criteria for such systems subject to two different boundary conditions by employing Gauss' divergence theorem and certain impulsive differential inequalities with delay.

Abstract : In this article, A new principle of geometric design for blade's surface of the impeller is provided. This is a optimal control problem for boundary geometric shape of flow and control variable is the surface of the blade. We give a minimum functional depending on geometry of the blade's surface and such that the flow's lossless achieve minimum. The existence of the solution of the optimal control problem is given and Euler-Lagrange equation for the surface of blade is derived.

^fSubsidized by the Special for Major State Basic Research Projects G1999032801 and NSFC Project 50136030

Abstract : We present the role of invariants in obtaining exact solutions of differential equations. Firstly, conserved vectors of a p.d.e. allow us to obtain reduced forms of the p.d.e. for which some of the Lie point symmetries are easily concluded and, therefore, providing a mechanism for further reduction. Secondly, invariants of reduced forms of a p.d.e. are obtainable from a variational principle even though the p.d.e. itself does not admit a Lagrangian. In this latter case, the reductions carry all the usual advantages regarding Noether symmetries and double reductions. The examples we consider are nonlinear evolution type equations like the Korteweg-deVries equation but a detailed analysis is made on the Fisher equation (which describes reaction- diffusion waves in biology, inter alia). Other diffusion type equations lend themselves well to the method we describe (e.g., the Fitzhugh-Nagumo equation).

Abstract : In this paper the author considers the Cauchy problem of the weakly damped driven long-wave-short-wave resonance equations. By making use of a Strichartz type inequality for the solution, decomposing suitably the solution semigroup into a decay part and a more regular part, and ruling out the ``vanishing" and ``dichotomy" of the solution he proves the existence of the global attractor and the asymptotic smoothing effect of the solutions.

Abstract : In this paper we prove the strong unique continuation property for a Lamé system with Lipschitz coefficients in the plane. The proof relies on reducing the Lamé system to a first order elliptic system and suitable Carleman estimates with singular weights.

Abstract : The original Anderson-Chaplain model was proposed in 2000 to explain how secondary tumors can remain undetected in the presence of the primary tumor yet suddenly appear upon surgical removal of the primary tumor. It is a nonlinear systems of parabolic PDES. Here we only consider the steady state solution, and thus the systems can be reduced to a nonlinear nonlocal elliptic PDES with two unknown. By a fixed point theorem it is shown that the solution exists, and is unique with an additional assumption.

Abstract : In this paper, the bifurcation method of planar systems and simulation method of differential equations are employed to investigate the bounded waves of the Camassa-Holm equation.

Abstract : The purpose of this paper is to study a regularization method of solutions of ill-posed problems involving hemivariational inequalities in Banach spaces.Under the assumption that the hemivariational inequality be solvable, a strongly convergent approximation procedure is designed by means of the so-called Browder-Tikhonov regularization method. Our results generalize and extend previously known theorems.

^fFinanced partially by: NNSF of China Grant No.10171008, NSF of Hunan Province Grant No.03JJY3003.

Abstract : In this paper, we study the energy decay rate for the thermoelastic Bresse system which describes the motion of a linear planar, shearable thermoelastic beam. The system consists of three one-dimensional wave equations and two one-dimensional heat equations which are coupled by the terms of displacement and temperature and their spatial derivatives. Exponential decay rate of energy is obtained for the case when longitudinal and vertical waves have the same speed. When the wave speeds are different, a polynomial type decay rate is obtained.

Abstract : Denote the infimal convolution and the supremal deconvolution of a function f by

Applying these results we can establish the comparison principle for viscosity super- and sub-solutions of the Dirichlet problem of degenerate prescribed curvature equations:

Previous work on this problem is also reviewed.

Abstract : In this paper the uniformly almost periodic functions in the sense of Bohr is generalized. we call them Besicovitch uniformly almost periodic functions. Some properties of this class of functions are discussed. Then the existence and uniqueness of Besicovitch almost periodic solution of wave equations involving reflection of the argument is investigated.

Abstract : Consider the existence of nontrivial solutions for the following biharmonic equation

Abstract : This paper is concerned with the Holling-Tanner prey-predator model with diffusion subject to the homogeneous Neumann boundary condition:

^f This work was supported by the National Natural Science Foundation of China 10471022, and the Ministry of Education of China Science and Technology Major Projects Grant 104090

Key words: Sturm-Liouville differential operators, dissipative operators, eigenfunctions, completeness, characteristic determinant.

Abstract : This paper introduces an efficient meshless computational scheme for analyzing the solution of Laplace equation in a domain containing a crack tip. The singularity at the crack tips could impair considerably the convergence and efficiency of the approximation method; thus, many classical mathematical methods cannot be applied directly to handle the problem of singularity. To overcome these difficulties, we employ a meshless approximation method derived from compactly supported radial basis functions (CSRBFs), which possesses a truly mesh free algorithm and a simple mathematical formulation. CSRBF is a class of continuously differentiable, positive definite and integrable functions, so it can easily be used to solve high order differential equations with boundary singularity and concave domains. To enhance the effective of the proposed scheme, the computational algorithm is incorporated with domain decomposition and least square approximation. Numerical examples show that the combined CSRBFs scheme and domain decomposition produces a high degree of accuracy and fast rate of convergence in the computations.

Abstract : In this talk, the result on the coexistence state of chemostat model with diffusion is introduced. Some simulations are also done to complement the mathematical analysis. The main ingredients include maximum principle, global bifurcation theory and fixed point index theory.

Abstract : We show an improved Hardy inequality and use the improved Hardy inequality and variational techniques to discuss the existence of nontrivial solution for following the weighted eigenvalue problem:

Abstract : We construct the fundamental solution for a general hyperbolic-parabolic system of conservation laws along a weak shock profile. Our formulation has explicit dependence on the shock strength. This allows us to perform nonlinear stability analysis for the shock wave, and obtain detailed asymptotic behavior of the solution. The result applies to compressible Navier-Stokes equations and the magnetohydrodynamics, even in the case of having multiple eigenvalues in the transversal fields.

Abstract : The concept of the derivative-dependent functional separable solution (DDFSS), as a generalization to the functional separable solution, is proposed. As an application, it is used to discuss the generalized nonlinear diffusion equations based on the generalized conditional symmetry approach. As a consequence, a complete list of canonical forms for such equations which admit the DDFSS is obtained and some exact solutions to the resulting equations are described.

Abstract : We consider the Navier-Stokes equations, a fundamental model in continuum mechanics, in W Í R³,

It is well-known that the weak solution to the 3-D Navier-Stokes equations exists globally for any given u₀ Î L²(W), but the uniqueness and regularity are still open and challenging problems. One direction of the regularity study is to find the sufficient conditions to guarantee the regularity (and uniqueness) of the weak solutions. Recently, we established the following regularity criteria, that is if one of the following conditions is satisfied, then the weak solution actually is strong.

A.   Any one component of the velocity field, say u₃, belongs to

B.   Gradient of any one component of the velocity field, say Ñu₃, belongs to

C.   The pressure P belongs to

D.   The domain W is bounded, and u and P satisfy

E.   Let w = Ñ×u be the vorticity field. The direction of the vorticity field

The above results are substantial improvement after the research works of a lot of mathematicians, such as P. Constantin, C. Fefferman, G. P. Galdi and J. Serrin. The estimates used in establishing the above regularity criteria are delicate and new.