四阶椭圆方程基态解的集中性态

该文分析了四阶椭圆方程△^２u=|x|^au^p-1,x∈Ω; u=\Delta u=0 , x ∈аΩ, (Ω表示Rⁿ中以原点为中心的球)基态解的集中性态,并证明了当p趋近于 2^*=\frac{2n}{n-4} (n>4)时基态解u_p集中在Ω的边界附近.     

拟齐性偏微分方程的多项式解

设p是Rⁿ上具C^∞系数的线性偏微分算子,关于拟相似变换δ_τ(x)=(τ>0)是m次拟齐性的,m>0,如果a₁,a₂,…,a_n全为正有理数或m∈M={α·a,α∈Iⁿ₊},则方程p[u]=0的多项式解空间必为无穷维的.     

非线性常微分方程初值问题的间断有限元

该文用m次间断有限元求解非线性常微分方程初值问题u'=f(x,u),u(0)=u₀,用单元正交投影及正交性质证明了当m≥1时,m次间断有限元在节点x_j的左极限U(x_j-0)有超收敛估计(u-U(x_j-0)=O(h^2m+1),在每个单元内的m+1阶特征点x_ji上有高一阶的超收敛性(u-U)(x_ji)=O(h^m+2).     

具有高阶非线性项的广义KdV方程的

研究具有高阶非线性项的广义KdV方程 u_t + a (1 + buⁿ)uⁿ u_x + u_xxx = 0, 这里n ≥1, a, b是实数且a ≠ 0. 用动力系统的定性理论和分支方法, 讨论了该方程的孤立波解的解析表达式和孤立波的分支, 并给出了孤立波的分支图, 解决了孤立波的存在性及其个数等问题.     

CH-γ方程的4类有界波解

最近,许多作者研究过下面的CH-γ方程 u_t+c₀ u_x+ 3uu_x-α²(u_xxt+ uu_xxx+2u_xu_xx)+γ u_xxx=0,其中α², c₀和γ是参数.在该方程的有界波研究中,已有的文献主要考虑α²>0的情形,对于α²<0的情形,Dullin等叙述了3种有界波(正常孤立波、紧孤立波和周期尖波)的存在性,但没有给出具体证明.在这篇文章中,主要考虑α²<0的情形,文中不仅证明4种有界波(周期波、广义紧孤立波、广义扭波和正常孤立波)的存在性,而且还给出了它们的显式表达式或隐式表达式.为验证其结果的正确性,文中还用计算机绘出了几组有界波解的图形以及它们的数值模拟图.     

求非线性问题多解的搜索延拓法

基于-Δ的特征系λ_j,φ_j,逼近非线性问题 Δu+f(u)=0(在Ω中), u=0(在¶Ω上)的多重解. 提出了一种新的搜索延拓法(SEM),它由三层子空间上的三种算法组成. 对f(u)=u³,在正方形和L形区域上完成了数值实验.这些结果表明,对应于-Δ的每个k重特征值, 至少存在3^k-1个不同的非零解(猜想1).     

脉冲泛函微分方程的渐近性态

该文讨论脉冲泛函微分方程$\left\{\begin{array}{ll}x^,(t)=f(t,x_t), t≥ t₀,△x=I_k(t,x(t^-)), t=t_k,k∈ Z⁺,给出了方程零解渐近稳定性和一致渐近稳定性的充分条件,指出这些条件推广或改进了文献[7--9]的相应结论.     

具投放的中立型时滞竞争扩散系统正周期解的存在性

利用Nussbaum 度理论建立了具投放的中立型时滞竞争扩散系统 x′₁(t)=x₁(t)[a₁(t)-b₁(t)x₁(t)-c₁(t)y(t) ]+D₁(t)[x₂(t)-x₁(t)]+S₁(t), x′₂(t)=x₂(t)[a₂(t)-b₂(t)x₂(t)]+D₂(t)[x₁(t)-x₂(t) ]+S₂(t), y′(t)=y(t)[a₃(t)-b₃(t)y(t)-α(t)y(t-τ₁(t))-β(t) ∫⁰_τ k(s)y(t+s)ds -γ(t)y’(t-τ₂(t))-c₃(t)x₁(t)]. 存在正周期解的一个充分条件.     

一类带权函数的拟线性椭圆方程

该文利用变分方法讨论了方程 -△_p u=λa(x)(u⁺)^p-1-μa(x)(u^-)^p-1+f(x, u), u∈W₀^1,p(\Omega)在(λ, μ)\not\in ∑_p和(λ, μ) ∈ ∑_p 两种情况下的可解性, 其中\Omega是 R^N(N≥3)中的有界光滑区域, ∑_p为方程 -△_p u=α a(x)(u⁺)^p-1-βa(x)(u^-)^p-1, u∈ W₀^1,p(\Omega)的Fucik谱, 权重函数a(x)∈ L^r(\Omega) (r≥ N/p)$且a(x)>0 a.e.于\Omega, f满足一定的条件.     

误差为鞅差序列的部分线性模型中估计的强相合性

考虑回归模型:y_i=x_{i ^β} +g(ti)+σ_ie_i ,i=1,2,...,n,其中 σ_i=f(u_i), (x_i,t_i,u_i)是固定非随机设计点列,f(^.),\ g(^.)$\ 是未知函数,β是待估参数,e_i是随机误差且关于非降σ -代数列{F_i,i≥1} 为鞅差序列.对文献[1]给出的基于f^(.)及g(^.)的一类非参数估计的β的最小二乘估计β_n和加权最小二乘估计β_n,在适当条件下证明了它们的强相合性,推广了文献[6]在e_i为iid情形下的结果.     

Blow up of solutions of a generalized Boussinesq equation

Consider the Cauchy problem u_tt = (f(u))_xx + u_xxtt x R, t 0, u(x,0) = u₀(x), u_t(x,0) = u₁(x),7rcub; where f : R R C, f(0) = ). After treatment of the local existenceproblem, we show the blow up of the solution of the equation(1) under the folowing assumptions. Let > 0 be real, such that 2(l + 2)F(u) uf(u), (v₀, Pv₀)_l² + _- F(u₀)dx < 0 where P = 1 - d²/dx², F'(s), and v₀ is given by u₁(x,0) = (v₀(x))_x. Then we focus on various perturbations of the question. We alsostudy the vectorial case in the same way, and finally we giveexamples.     

Asymptotic simplification for a reaction-diffusion problem with a nonlinear boundary condition

We study non-negative solutions of the porous medium equationwith a source and a nonlinear flux boundary condition, u_t =(u^m)_xx + u^p in (0, ), x (0, T); – (u^m)_x (0, t) = u^q (0,t) for t (0, T); u (x, 0) = u₀ (x) in (0, ), where m > 1,p, q > 0 are parameters. For every fixed m we prove thatthere are two critical curves in the (p, q-plane: (i) the criticalexistence curve, separating the region where every solutionis global from the region where there exist blowing-up solutions,and (ii) the Fujita curve, separating a region of parametersin which all solutions blow up from a region where both globalin time solutions and blowing-up solutions exist. In the caseof blow up we find the blow-up rates, the blow-up sets and theblow-up profiles, showing that there is a phenomenon of asymptoticsimplification. If 2q < p + m the asymptotics are governedby the source term. On the other hand, if 2q > p + m theevolution close to blow up is ruled by the boundary flux. If2q = p + m both terms are of the same order.     

Algebraic decay and variable speeds in wavefront solutions of a scalar reaction-diffusion equation

Wavefront solutions of scalar reaction-diffusion equations havebeen intensively studied for many years. There are two basiccases, typified by quadratic and cubic kinetics. An intermediatecase is considered in this paper, namely, u_l = u_xx + u²(1 –u). It is shown that there is a unique travelling-wave solution,with a speed 1/2, for which the decay to zero ahead of the waveis exponential with x. Moreover, numerical evidence is presentedwhich suggests that initial conditions with such exponentialdecay evolve to this travelling-wave solution, independentlyof the half-life of the initial decay. It is then shown thatfor all speeds greater than 1/2 there is also a travelling-wavesolution, but that these faster waves decay to zero algebraically,in proportion to 1/x. The numerical evidence suggests that initialconditions with such a decay rate evolve to one of these fasterwaves; the particular speed depends in a simple way on the detailsof the initial condition. Finally, initial conditions decayingalgebraically for a power law other than 1/x are considered.It is shown numerically that such initial conditions evolveeither to an algebraically decaying travelling wave, or in somecases to a wavefront whose shape and speed vary as a functionof time. This variation is monotonic and can be quite pronounced,and the speed is a function of u as well as of time. Using asimple linearization argument, an approximate formula is derivedfor the wave speed which compares extremely well with the numericalresults. Finally, the extension of the results to the more generalcase of u_l = u_xx + u^m(1 – u), with m > 1, is discussed.     

Adaptive numerical schemes for a parabolic problem with blow-up

In this paper we present adaptive procedures for the numericalstudy of positive solutions of the following problem: u_t = u_xx (x, t) (0, 1) x [0, T), u_x(0, t) = 0 t [0, T), u_x(1, t) = u^p(1, t) t [0, T), u(x, 0) = u₀(x) x (0, 1), with p > 1. We describe two methods. The first one refinesthe mesh in the region where the solution becomes bigger ina precise way that allows us to recover the blow-up rate andthe blow-up set of the continuous problem. The second one combinesthe ideas used in the first one with moving mesh methods andmoves the last points when necessary. This scheme also recoversthe blow-up rate and set. Finally, we present numerical experimentsto illustrate the behaviour of both methods.     

The determination of a coefficient in a parabolic equation from input sources

The authors consider the question of recovering the coefficientq from the equation u_t – u_xx + q(x)u = f_j(x) with homogeneousinitial and boundary conditions. The nonhomogeneous source terms form a basis for L²(0,1).It is proved that a unique determination is possible from datameasurements consisting of either the flux at one end of thebar or the net flux leaving the bar, taken at a single fixedtime for each input source. An algorithm that allows efficientnumerical reconstruction of q(x) from finite data is given.     

Smoothness Properties of the Unit Ball in a JB*-Triple

An element a of norm one in a JB^*-triple A is said to be smoothif there exists a unique element x in the unit ball A₁^* of thedual A^* of A at which a attains its norm, and is said to beFréchet-smooth if, in addition, any sequence (x_n) ofelements in A₁^* for which (x_n(a)) converges to one necessarilyconverges in norm to x. The sequence (a²ⁿ⁺¹) of odd powers ofa converges in the weak^*-topology to a tripotent u(a) in theJBW^*-envelope A^** of A. It is shown that a is smooth if andonly if u(a) is a minimal tripotent in A^** and a is Fréchet-smoothif and only if, in addition, u(a) lies in A.     

Oscillation Properties of a Semi-discrete Approximation to the Beam Equation with a Second Order Term

The solution of the equation w(x)u_tt+[p(x)u_xx]_xx–[p(x)u_x]_x=0, 0< x < L, t > 0, where it is assumed that w, p,and q are positive on the interval [0, L], is approximated bythe method of straight lines. The resulting approximation isa linear system of differential equations with coefficient matrixS. The matrix S is studied under a variety of boundary conditionswhich result in a conservative system. In all cases the matrixS is shown to be similar to an oscillation matrix.