ON THE ATTRACTOR FOR A SEMILINEAR WAVE EQUATION WITH CRITICAL EXPONENT AND NONLINEAR BOUNDARY DISSIPATION

ABSTRACT

Long time behavior of a semilinear wave equation with nonlinear boundary dissipation and critical exponent is considered. It is shown that weak solutions generated by the wave dynamics converge asymptotically to a global and compact attractor. In addition, regularity and structure of the attractor are discussed in the paper. While this type of results are known for wave dynamics with interior dissipation this is, to our best knowledge, first result pertaining to boundary and nonlinear dissipation in the context of global attractors and their properties.     

DECAY RATES TOWARD STATIONARY WAVES OF SOLUTIONS FOR DAMPED WAVE EQUATIONS

Abstract This paper is concerned with the initial-boundary value problem for damped wave equations with a nonlinear convection term in the half space R＋{utt-txx＋ut＋f（u）x=0,t〉0,x∈R＋,u（0,x）=u0（x）→u＋,asx→＋∞,ut（0,x）=u1（x）,u（t,0）=ub.For the non-degenerate case f]（u＋） 〈 0, it is shown in [1] that the above initialboundary value problem admits a unique global solution u（t,x） which converges to the stationary wave φ（x） uniformly in x ∈ R＋ as time tends to infinity provided that the initial perturbation and/or the strength of the stationary wave are sufficiently small. Moreover, by using the space-time weighted energy method initiated by Kawashima and Matsumura [2], the convergence rates （including the algebraic convergence rate and the exponential convergence rate） of u（t, x） toward φ（x） are also obtained in [1]. We note, however, that the analysis in [1] relies heavily on the assumption that f＇（ub） 〈 0. The main purpose of this paper is devoted to discussing the case of f＇（ub）= 0 and we show that similar results still hold for such a case. Our analysis is based on some delicate energy estimates.     

MULTIPLE AND SIGN-CHANGING SOLUTIONS FOR NONLINEAR ELLIPTIC EQUATION WITH CRITICAL POTENTIAL AND CRITICAL PARAMETER

Some embedding inequalities in Hardy-Sobolev space are proved.Furthermore,by the improved inequalities and the linking theorem,in a new k-order Sobolev-Hardy space,we obtain the existence of sign-changing solutions for the nonlinear elliptic equation {-△(k)u:=-△u-(((N-2)2)/4)U/︱X︱2-1/4 sum from i=1 to(k-1) u/(︱x︱2(In(i)R/︱x︱2))=f(x,u),x ∈Ω,u=0,x ∈Ω,where 0 ∈ΩBa(0)RN,N≥3,ln(i)=i éj=1 ln(j),and R=ae(k-1),where e(0)=1,e(j) = ee(j-1) for j≥1,ln(1)=ln,ln(j)=ln ln(j-1) for j≥2.Besides,positive andnegative solutions are obtained by a variant mountain pass theorem.     

小展弦比波的Green-Naghdi渐进模型的行波解

本文研究了小展弦比波的Green-Naghdi渐进模型. 利用平面自治系统的稳定性分析方法, 在不同的参数条件下, 讨论了它的行波系统的分岔并且给出了对应的相图, 得到了光滑周期波解, 广义扭波解, 广义反扭波解, 广义紧波解, 周期尖波解, 孤波解和孤立尖波解的精确表达式. 进一步, 通过数学软件Maple模拟了这些解.     

CRITICAL EXPONENTS AND CRITICAL DIMENSIONS FOR NONLINEAR ELLIPTIC PROBLEMS WITH SINGULAR COEFFICIENTS

Let B1 ■ RNbe a unit ball centered at the origin. The main purpose of this paper is to discuss the critical dimension phenomenon for radial solutions of the following quasilinear elliptic problem involving critical Sobolev exponent and singular coefficients:-div(|▽u|p-2▽u) = |x|s|u|p*(s)-2u + λ|x|t|u|p-2u, x ∈ B1,u|■B1= 0,where t, s -p, 2 ≤ p N, p*(s) =(N+s)p N-pand λ is a real parameter. We show particularly that the above problem exists infinitely many radial solutions if the space dimension N p(p- 1)t + p(p2- p + 1) and λ∈(0, λ1,t), where λ1,t is the first eigenvalue of-△p with the Dirichlet boundary condition. Meanwhile, the nonexistence of sign-changing radial solutions is proved if the space dimension N ≤(ps+p) min{1,p+t p+s}+p2p-(p-1) min{1,p+t p+s}and λ 0 is small.     

BOUNDING PYRAMIDS AND BOUNDING CONES FOR TRIANGULAR BEZIER SURFACES

1. IntroductionA 6ounding pyramid/cone of a rational B6zier surface patch is a pyramid/cone whichhas the property that if its vertex is translated to any point on the B6zier patch, thepatch will lie completely outside the pyramid/cone. These kinds of pyramids/conesare useful tools in detecting closed loops in surface/surface intersections[2, 3] and determining directions for which a surface is single valued[5]. While methods of finding bounding pyramids and bounding cones for rectangular B6zi…     

深水表面波可积发展方程的行波解与分支

本文研究了small-aspect-ratio波方程和深水表面波可积发展方程的行波解问题.利用微分方程定性理论的方法,分析了行波系统的相图分支,获得了孤立波解的精确表达式.     

高阶C-B样条曲线曲面的显式形式

1引言基于多项式空间span{1,t,t~2,…,t~k}的B样条和Bézier曲线(面)是构造自由曲线、曲面强有力的工具,但是它们不能精确表示圆弧、椭圆等,也不能精确地表示正弦曲线和二次曲面,于是文献[1]提出了一种新的三次曲线(面)模型,称为C曲线(面),它们是低次多项式样条曲线(面)的拓广,具有很多B样条的良好性质,如对称性、保凸性等,不仅能精确表示二次曲线和曲面和某些超越曲线,而且克服了NURBS求导求积复杂的困难,因此引起了国内外广泛的关注,近年来涌现了大量的文献.文[5]将其三次C-B样条推广到了高阶的情形,给出了任意阶C-B样条曲线(面)的积分递推公式,并发展了许多诸     

非线性波动方程的弱隐式与显式差分方法

广泛出现于物理、化学、机械动力学、生物、几何学等领域的非线性波动方程已经有很多的研究工作,Sine-Gordon方程和非线性受迫振动方程就是典型的例子.周毓麟教授在[1]中研究了非线性波动方程组     

ESTIMATE OF THE UPPER CRITICAL FIELD AND CONCENTRATION FOR SUPERCONDUCTOR

The effect of an applied magnetic field on an inhomogeneous superconductor is studied and the value of the upper critical magnetic field Hc3 at which superconductivity can nucleate is estimated. In addition, the authors locate the concentration of the order parameter, which depends on the inhomogeneous term a(x). Unlikely to the homogeneous case, the order parameter may concentrate in the interior of the superconducting material, due to the influence of the inhomogeneous term a(x).     

DECAY ESTIMATES OF PLANAR STATIONARY WAVES FOR DAMED WAVE EQUATIONS WITH NONLINEAR CONVECTION IN MULTI-DIMENSIONAL HALF SPACE

This paper is concerned with the initial-boundary value problem for damped wave equations with a nonlinear convection term in the multi-dimensional half space R n + : u tt u + u t + divf (u) = 0, t > 0, x = (x 1 , x ′ ) ∈ R n + := R + × R n 1 , u(0, x) = u 0 (x) → u + , as x 1 → + ∞ , u t (0, x) = u 1 (x), u(t, 0, x ′ ) = u b , x ′ = (x 2 , x 3 , ··· , x n ) ∈ R n 1 . (I) For the non-degenerate case f ′ 1 (u + ) < 0, it was shown in [10] that the above initialboundary value problem (I) admits a unique global solution u(t, x) which converges to the corresponding planar stationary wave φ(x 1 ) uniformly in x 1 ∈ R + as time tends to infinity provided that the initial perturbation and/or the strength of the stationary wave are sufficiently small. And in [10] Ueda, Nakamura, and Kawashima proved the algebraic decay estimates of the tangential derivatives of the solution u(t, x) for t → + ∞ by using the space-time weighted energy method initiated by Kawashima and Matsumura [5] and improved by Nishihkawa [7]. Moreover, by using the same weighted energy method, an additional algebraic convergence rate in the normal direction was obtained by assuming that the initial perturbation decays algebraically. We note, however, that the analysis in [10] relies heavily on the assumption that f ′ (u) < 0. The main purpose of this paper isdevoted to discussing the case of f ′ 1 (u b ) ≥ 0 and we show that similar results still hold for such a case. Our analysis is based on some delicate energy estimates.     

Bzier曲线和曲面包围的面积和体积算法

§1.引言 曲线、曲面造型和立体造型的理论和方法是迅速发展的 CAD/CAM技术的重要基础之一.在造船、航空、汽车、模具、机械和建筑等行业的CAD系统中,经常需要计算平面上闭曲线包围的面积以及三维立体图形的表面积和体积.     

SOLITARY WAVE AND PERIODIC WAVE SOLUTIONS FOR THE NON-NEWTONIAN FILTRATION EQUATIONS WITH NONLINEAR SOURCES AND A TIME-VARYING DELAY

This paper is concerned with the non-Newtonian filtration equations with nonlinear sources and a time-varying delay. By an extension of Mawhin's continuation theorem and some analysis methods,several sufficient conditions ensuring the existence of solitary wave and periodic wave solutions are obtained. Some corresponding results in the literature are improved and extended. An example is given to illustrate the effectiveness of our results.     

LOCAL THEORY IN CRITICAL SPACES FOR COMPRESSIBLE VISCOUS AND HEAT-CONDUCTIVE GASES

We are concerned with local existence and uniqueness of solutions for a general model of viscous and heat-conductive gases with low regularity assumptions on the initial data (the velocity and the temperature may be discontinuous). Local well-posedness is showed to hold in spaces which are critical with respect to the scaling of the equations, provided that the initial density is close enough to a positive constant. When initial data are a trifle more regular, local well-posedness holds for any initial density bounded away from zero. This former result lies on new estimates for linear heat equations with a non constant diffusion coefficient.     

LOCAL WELL-POSEDNESS OF INTERACTION EQUATIONS FOR SHORT AND LONG DISPERSIVE WAVES

The well-posedness of the Cauchy problem for the system{iδtu+δx^2u=uv+|u|^2u,t,x∈IR，δtv+δxHδxv=δx|u|^2,u(0,x)=u0(x),v(0,x)=v0(x),is considered. It is proved that there exists a unique local solution (u(x,t), v(x,t))∈C（[0,T);H^s)&#215;C([0,T);Hs^-1/2) for any initial data (u0,v0)∈H^s(IR)&#215;H^s-1/2(IR)(s≥1/4) and the solution depends continuously on the initial data.