Session 9: Partial Differential Equations On the Classification of Radial Solutions of some Biharmonic Equations

Jann-Long Chern* and Zhi-You Chen
Department of Mathematics
National Central University
Chung-Li 320, Taiwan
chern@math.ncu.edu.tw

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




Oscillation Criteria for Impulsive Parabolic Differential Systems with Delay

Xilin Fu
Department of Mathematics
Shandong Normal University
Jinan, Shandong 250014, P.R. China
xilinfu@beelink.com

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.




Optimal Design for Geometry of Blade and Boundary Shape Control of Navier-Stokes Equationsf

Aixiang Huang
School of Science
Xi'an Jiaotong University
Xi'an, 710049, P.R. China
axhuang@mail.xjtu.edu.cn

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




Invariants of Nonlinear Evolution Type Equations and their Exact Solutions

Abdul Hamid Kara
School of Mathematics
Wits University
Private Bag 3
Wits, 2050, South Africa
kara@maths.wits.ac.za

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).




Long Time Behavior for the Weakly Damped Driven Long-Wave-Short-Wave Resonance Equations

Yong-Sheng Li
School of Mathematics
South China University of Technology
Guangzhou, Guangdong 510640, P. R. China
liys_ scut@mail.edu.cn

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.




Strong Unique Continuation for the Lamé System with Lipschitz Coefficients

Ching-Lung Lin*
Department of Mathematics
National Chung Cheng University
Chia-Yi 62117, Taiwan
Email:cllin@math.ccu.edu.tw

Jenn-Nan Wang
Department of Mathematics
National Taiwan University
Taipei 106, Taiwan
jnwang@math.ntu.edu.tw

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.




The Steady-State Anderson-Chaplain Chemotaxtic Model

Yanping Lin
Department of Mathematics
The University of Alberta
Edmonton, Alberta T6G 2G1
ylin@math.ualberta.ca

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.




Two New Types of Bounded Waves of Camassa-Holm Equation

Zheng-Rong Liu
School of Mathematical Sciences,
South China University of Technology,
Guangzhou, 510640, P.R.China
liuzhr@scut.edu.cn

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.

ut+2kux-uxxt+3uux = 2uxuxx+uuxxx
Two new types of bounded waves are found and their implicit expressions are obtained. Both qualitative and numerical results show that they possess some properties of compactons or kink waves. Therefore they are called compacton-like and kink-like wave respectively.




Strong Convergence Results for Hemivariational Inequalitiesf

Zhenhai Liu
Department of Mathematics
Changsha University of Science and Technology
Hunan 410077, P.R. China
liuzhenh@cscu.edu.cn

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.




Energy Decay Rate of the Thermoelastic Bresse System

Zhuangyi Liu
Department of Mathematics and Statistics
University of Minnesota at Duluth
Duluth, MN 55812, USA
zliu@d.umn.edu

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.




An Application of C1,1 Approximation to Comparison Principles for Viscosity Solutions of Curvature Equations

Yousong Luo*
School of Mathematical
RMIT University
Melbourne, Vic 3001, Australia
yluo@rmit.edu.au

Andrew Eberhard
Geospatial Sciences
RMIT University
Melbourne, Vic 3001, Australia
andy.eb@rmit.edu.au

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

fl(x): =
inf
u \Bbb R n 


f(u)+ 1
2l
||x-u||2

    and     fl(x): =
sup
u \Bbb R n 


f(u)- 1
2l
||x-u||2

respectively. By C1,1 approximation we mean the double envelope of Lasry-Lions type:
fl\mid m: = hm( fl)
for 0 < m < l < [`(l)], where hmf(x) = ( fm) m(x) and [`(l)] is a constant associated with f. It is well known that fl\mid m is C1,1( \Bbb R n) . In a recent study of such an envelope we have obtained some differential properties of fl\mid m, such as the convergence of the sub-jets (i.e. the sub-differential together with the sub-Hessian) of fl\mid m to those of f, and boundedness of these sub-jets.

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

F[u] = F(Du, D2u) = F(k1, kn) = 0      in     W,        u(x) = f(x)      on     W
where k's are the principal curvatures of the graph of u and
F(k) = Sk,l(k) = Sk(k)
Sl(k)
       for        0 l < k n
where
Sk(k) =

i1 < < ik 
ki1 ki2 kik        for        1 k n        and        S0(k) = 1.

Previous work on this problem is also reviewed.




Besicovitch Almost Periodic Solutions for Wave Equations Involving Reflection of the Argument

Daxiong Piao* and Hongyan Zhou
Department of Mathematics
Ocean University of China
Qingdao, 266071, China
davidpiao@yahoo.com  and   bshzhhy@sohu.com

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.




Nonlinear Biharmonic Equations in R4 with Critical Potential

Yao-Tian Shen*
School of Mathematical Sciences
South China University of Technology
Guangzhou 510640, China
maytshen@scut.edu.cn

Zhi-Hui Chen
School of Mathematical Sciences
South China University of Technology
Guangzhou 510640, China
mazhchen@scut.edu.cn

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







D2 u = m u
|x|4ln2 R/|x|
+f(x,u),
x W,
u = u
n
= 0,
x W,
where W BR(0) R4 is a bounded domain including the origin, m R, n is the unit outer normal to W, f has subcritical growth (with respect to t), that is, for all a > 0,

lim
t 
f(x,t)
ea|t|4/3
= 0.
We establish a Hardy-type inequality in H01(W), and show that the best constant is 1. Then we know the critical potential, related to biharmonic equation in R4, is [1/( |x|4ln2 R/|x|)]. By using the Eklend's variational principle, we discuss the eigenvalue problem of biharmonic equation with critial potential, and then the existence of many solutions of the above problem.




Positive Steady-states of the Holling-Tanner Prey-predator Model with Diffusionf

Mingxin Wang
Department of Mathematics
Xuzhou Normal University
Xuzhou 221116, P. R. China Department of Mathematics
Southeast University
Nanjing 210018, P. R. China
(This is a joint work with Dr. Rui Peng)
mxwang@@seu.edu.cn

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







ut - d1 Du = au - u2 -\dfracuvm+u
inW×(0, ),
vt - d2 Dv = bv - \dfracv2gu
in W×(0, ),
h u = h v = 0
on W×(0, ),
where parameters are all positive constants and W \Bbb RN is a bounded domain with smooth boundary W. With the more accurate priori estimates of the lower and upper bounds of positive steady-states, we obtain the existence, bifurcation and non-existence of positive non-constant steady-states.

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




Dissipativeness of Non-Self-Adjoint Sturm-Liuville Operators And Completeness of Their Eigenfunctions

Zhong Wang*
Department of Mathematics
ZhaoQing University
ZhaoQing GuangDong P.R.C. 526061, China
kyczwang@zqu.edu.cn

Hongyou Wu
Department of Mathematics
Northern Illinois
University, DEKALB, Illinois 60115-2854, U.S.A.
wu@math.niu.edu

Abstract

In this paper, non-self-adjoint Sturm-Liuville operators in Weyl's limit-circle case are studied. We first determine all the non-self-adjoint boundary conditions yielding dissipative operators for each allowed Sturm-Liuville differential expression. Then, using the characteristic determinant, the completeness of the system of eigenfunctions and associated functions for these dissipative operators is proved.

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




A Mesh-free Domain Decomposition Scheme for Crack Tip Analysis

S.M. Wong* and T.S. Li
School of Science and Technology
The Open University of Hong Kong
30 Good Shepherd Street, Homantin, Kowloon, Hong Kong
anwong@ouhk.edu.hk

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.




Coexistence of Chemostat Model with Diffusion

Jianhua Wu
College of Mathematics and Information Science
Shaanxi Normal University
Xi'an, 710062, P. R. China
jianhuaw@snnu.edu.cn
Joint with G.Wolkowicz and H.Nie

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.




Biharmonic Equation and An Improved Hardy Inequality

Yangxin Yao*
School of Mathematics
South China University of Technology
Guangzhou, Guangdong 510640, P. R. China
mayxyao@scut.edu.cn

Yaotian Shen
School of Mathematics
South China University of Technology
Guangzhou, Guangdong 510640, P. R. China
maytshen@scut.edu.cn

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:







D2 u -m u
|x|4
= lu f(x),   
x W,
u = u
n
= 0,    
x W.




Fundamental Solutions of Hyperbolic-Parabolic Systems and Shock Wave Stability

Tai-Ping Liu
Department of Mathematics
Stanford University
Stanford, CA 94305, USA

Yanni Zeng*
Department of Mathematics
University of Alabama at Birmingham
Birmingham, AL 35294, USA
zeng@math.uab.edu

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.




New Variable Separation Approach: Application to Nonlinear Diffusion Equations

Shun-Li Zhang
Department of Mathematics
Northwest University
Xi'an 710069, People's Republic of China
zhang shunli@126.com

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.




Regularity Criteria for the Incompressible Navier-Stokes Systems

Yong Zhou
Department of Mathematics
Xiamen University
Shamen, Fujian 361005, China
Zhou_Yong@alumni.cuhk.net

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













u
t
+u ·u+P = Du,           in  W×(0,T)
div u = 0,                                 in  W×(0,T)
u = 0,                                    on  W×(0,T)
u(x,0) = u0(x),                      in  W
where u = u(x,t) R3 is the velocity field, P(x,t) is a scalar pressure, and u0(x) with divu0 = 0 in the sense of distribution is the initial velocity field.

It is well-known that the weak solution to the 3-D Navier-Stokes equations exists globally for any given u0 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

La(0,T;Lg(W)),    with    2
a
+ 3
g
1
2
,     6 < g .

B.   Gradient of any one component of the velocity field, say u3, belongs to

La(0,T;Lg(W)),    with    2
a
+ 3
g
3
2
,     3 < g .

C.   The pressure P belongs to

La(0,T;Lg(R3)),    with    2
a
+ 3
g
2,      3
2
< g .
or the gradient of the pressure P belongs to
La(0,T;Lg(R3)),    with    2
a
+ 3
g
3,     1 < g .

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

P
1+|u|d
La,g,     with    2
a
+ 3
g
= 5
2
- 3
2
d,     6
5-3d
< g ,     or
P
1+|u|d
La,g,     with    2
a
+ 3
g
= 7
2
- 3
2
d,     6
7-3d
< g .

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

x(x) = w(x)
|w(x)|
             is 1/2-Hölder continuous.

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.


File translated from TEX by TTH, version 2.00.
On 08 Dec 2004, 11:27.