二阶含阻尼项可化为“积分小”系数的微分方程解的振动性质

本文考虑下列二阶微分方程 (r(t)x′(t))′ q(t)x′(t) p(t)x(t)=0. (1) 和 (r(t)x′(t))′ q(t)x′(t) p(t)f(x(g(t)))=0 (2)解的振动性质。我们给出了方程(1)非振动解存在的充要条件和方程(2)存在振动解的充分判据。     

具有变指数反应项的多孔介质方程的爆破

本文研究下述具有变指数反应项的多孔介质方程解的爆破和整体存在性问题,u_t=?u~m+u~(p(x)),(x,t)∈?×(0,T),其中?为有界域或全空间R~N,p(x)为定义在?上满足条件0p_=infp(x)≤p(x)≤p+=supp(x)∞的连续函数.这个方程由于变指数p(x)与定义域?的空间结构之间的相互作用表现出丰富而有趣的动力学特性.粗略地讲,对于全空间R~N上的初值问题,如果p(x)≤1,则方程的解可能不具有唯一性,此时所有非平凡解均整体存在;如果p+m,此时一定存在爆破解.进一步,当1p(x)≤m+2/N时,所有非平凡解均爆破;当p(x)m+2/N时,存在非平凡整体解.当p_m+2/N时,本文构造的例子表明,对于某些p(x)所有非平凡解均爆破;而对于另外一些p(x),则可能存在整体非平凡解.在有界域上解的性质与全空间又有所不同.此时有p(x)和m及区域性质三个因素相互作用,而仅有一个临界指标p=m表征解的爆破行为.若p+m,则此时如同全空间情形存在爆破解;若p+m,则方程所有解均整体存在;又若p(x)m或者区域足够小,则方程存在整体解.最有意思的是,对于某些满足条件p_mp+的p(x),作者发现了对于这类方程特有的有界域上的Fujita现象.     

Diophantine方程y~2=px(x~2+2)

设p是大于3的奇素数.本文证明了:当p≡5或7(mod 8)时,方程y~2=px(x~2+2)无正整数解(x,y);当p≡1(mod 8)时,该方程至多有1组解;当p≡3(mod 8)时,该方程至多有2组解.     

关于一种近似牛顿法的收敛性

§1.前言 设X和Y是Banach空间,p(x)是定义在区域G X上并取值于Y的非线性算子。假定p(x)有Frechet导算子p’(x),为了近似解算子方程 p(x)=0, (1)研究了如下的迭代程序: x_(n 1)=x_n-A_np(x_n), A_(n 1)=2A_n-A_np(x_(n 1)A_n,(2)这里x_0∈G和A_0∈(Y→X)都是初始近似,其中x_0是方程(1)的近似解,而A_0则是p(x_0)的近似过算子。[1]在一些条件下证明了程序(2)收敛于方程(1)的解。     

二阶非线性阻尼常微分方程的振动性定理

考虑二阶非线性阻尼微分方程(α(t)ψ(x(t))x′(t))′ p(t)x′(t) q(t)f(x(t))=0 (1)和二阶非线性微分不等式x(t){(α(t)ψ(x(t))x′(t))′ p(t)x′(t) q(t)f(x(t))}≤0,(2)其中α,p,q∈C([t_0,∞)→(-∞,∞)),ψ,f∈C(R→R),并且α(t)>0,xf(x)>0 (x≠0).此外,我们总假设方程(1)的每一个解 x(t)可以延拓于[t_0, ∞)上.在任何无穷区间[T,∞)上,x(t)不恒等于零,这样的解叫正则解.一个正则解,若它有任意大的零点,则称为振动的;否则就称为非振动的.若方程(1)的所有正则解是振动的,则称方程(1)是振动的.关于不等式(2)的振动性的定义,与方程(1)的振动性的定义完全类似,不再赘述.     

一类非线性斯图谟-刘维尔方程两点边值问题解的存在性

本文讨论了如下非线性斯图谟-刘维尔方程的第一边值问题{p(x)u"(x)+f(u(x))=0,0相似文献   

可积的Riccati微分方程的不变量变换讨论

对于可积的Riccati微分方程:L[y]=-y′+p(x)yn+Q(x)y+R(x)(p(x)R(x)≠0,n≠0,1)(0)L[y]=-y′+p(x)y2+Q(x)y+R(x)(p(x)R(x)≠0)(1)利用其不变量变换,给出方程(0)和(1)的可积充分条件,并对方程(1)的特解形式L[y0]=0,讨论其不变量变换的等效性;同时,对方程(1)的非特解形式L[y0]≠0,讨论其可积性.     

一类具时滞耗散型Duffing方程的周期解

利用Mawhin重合度理论研究了一类耗散型时滞Duffing方程ax″+f[x′(t-τ1(t))]+cx+g(x(t-τ2(t)))=p(t)周期解的存在性,得到了该方程2π周期解存在的充分性定理.     

关于指数Diophantine方程x~2=D~(2m)-D~mp~n+p~(2n)

设D 1是正整数,p是适合p?D的素数.本文研究了指数Diophantine方程x~2=D~(2m)-D~mp~n+p~(2n)的满足m 1的正整数解.根据Diophantine方程的性质,结合已有的结论,运用初等方法确定了方程满足m 1的所有正整数解(D,p,x,m,n).这个结果修正并完整解决了文献[4]的猜想.     

不定方程x4-y4=n的整数解之我见

文 [1]对不定方程　　　　　 x4- y4=n (1)的整数解求法作了探讨 ,笔者认为有必要作一些说明 .容易验证 :奇数的四次方除以 16余 1.n =(x - y) (x +y) (x2 +y2 ) ,n(n >1)必为合数 ;若 (x,y)满足方程 (1) ,则(± x,± y)也满足方程 (1) ,故仅需考虑正整数解 .容易得到 (以下字母为正整数 ) :定理 1　 n =a2 ,2 a2 ,pa2 (p为素数 ,p≡3(mod8) )时 ,方程 (1)无正整数解 [2 ] .定理 2　方程 (1)有正整数解的充要条件是 n =PQ(P 相似文献   

用椭圆描述的四阶边值问题的两参数非共振条件

该文讨论四阶常微分方程边值问题u(4)=f(t,u,u″),0≤t≤1,u(0)=u(1)=u″(0)=u″(1)=0解的存在性,其中f:[0,1]×R×R→R连续.文中提出了一个保证该问题解存在的两参数非共振条件,该条件是用椭圆描述的.     

一类非线性波方程初边值问题解的爆破

该文研究如下具有非线性阻尼项和非线性源项的波方程的初边值问题 u_tt -u_xxt -u_xx -(σ(u²_x)u_x)_x+δ|u_t|^p-1u_t=μ|u|^q-1u, 0 < x <1, 0≤ t ≤T, (0.1) u(0, t)=u(1, t)=0, 0≤t≤ T, (0.2) u(x, 0)=u₀(x), u_t(x, 0)=u₁(x),0≤x≤1.(0.3) 文章将给出问题(0.1)--(0.3)的解在有限时刻爆破的充分条件, 同时将证明问题的局部广义解和局部古典解的存在性和唯一性.     

Long time classical solutions of a nonlinear wave equation

The existence and uniqueness of long time classical solutions of the Cauchy problem u_{t t}+μu_t = div(a(u)▽u), where a(u) = 1+u and μ ≥ 0, are studied for the case of two space dimensions. Let the initial data u(0,.) = φ and u_t(0,.) = ψ be supported compactly on R². Then for every T > 0, such a solution exists on [0,T] whenever (φ,ψ) is small enough in H⁴ (R²) x H³(R²). A result on the asymptotic relation between the maximal T and the size of the initial data is given.     

Scattering of small solutions to generalized benjamin-bona-mahony equation in several space dimensions

We show that the supremum norm of solutions with small initial data of the generalized Benjamin-Bona-Mahony equation u_t-△u_t=(b,▽u)+u^p(a,▽u)in x?Rⁿ,n≥2, with integer p≥3 , decays to zero like t^-2/3 if n=2 and like t^-1+6, for any δ0, if n≥3, when t tends to infinity. The proofs of these results are based on an analysis of the linear equation u_t-△=(b,▽u)) and the associated oscillatory integral which may have nonisolated stationary points of the phase function.     

Existence results for quasilinear elliptic equations with multivalued nonlinear terms

In this paper we study the existence of solutions u ∈ \({{W}^{1,p}_{0}}\) (Ω) with △ _p u ∈ L ²(Ω) for the Dirichlet problem 1 $$ \left\{ \begin{array} [c]{l}-\triangle_{p}u\left( x\right) \in-\partial{\Phi}\left( u\left( x\right) \right) +G\left( x,u\left( x\right) \right) ,x\in{\Omega},\\ u\mid_{\partial{\Omega}}=0, \end{array} \right. $$ where Ω ? R ^N is a bounded open set with boundary ?Ω, △ _p stands for the p?Laplace differential operator, ?Φ denotes the subdifferential (in the sense of convex analysis) of a proper convex and lower semicontinuous function Φ and G : Ω × R → 2^R is a multivalued map. We prove two existence results: the first one deals with the case where the multivalued map u ? G(x, u) is upper semicontinuous with closed convex values and the second one deals with the case when u ? G(x, u) is lower semicontinuous with closed (not necessarily convex) values.     

带权的 p-Laplacian 非线性特征值问题解的存在性

文利用变分方法讨论了方程-△p^u=λ a(x)(u^{+})^q-1-μ a(x)(u^-)^q-1+f(x,u), u∈W₀^1,p(Ω), 当 p≠q时的可解性. 其中Ω是 R^N(N≥ 3)中的有界光滑区域,权重函数a(x)∈ L^r(Ω), (r≥Np/Np-Nq+pq)且a(x)>0, a.e.于Ω, f满足某些条件.     

Existence of positive solutions to nonlinear elliptic equations involving critical Sobolev exponents

In this paper we extend the results of Brezis and Nirenberg in [4] to the problem $$\left\{ \begin{gathered} Lu = - D_i (a_{ij} (x)D_j u) = b(x)u^p + f(x,u) in\Omega , \hfill \\ p = (n + 2)/(n - 2) \hfill \\ u > 0 in\Omega , u = 0 \partial \Omega , \hfill \\ \end{gathered} \right.$$ whereL is a uniformly elliptic operator,b(x)≥0,f(x,u) is a lower order perturbation ofu ^p at infinity. The existence of solutions to (A) is strongly dependent on the behaviour ofa _ij (x), b(x) andf(x, u). For example, for any bounded smooth domain Ω, we have \(a_{ij} \left( x \right) \in C\left( {\bar \Omega } \right)\) such thatLu=u ^p possesses a positive solution inH ₀ ¹ (Ω). We also prove the existence of nonradial solutions to the problem ?Δu=f(|x|, u) in Ω,u>0 in Ωu=0 on ?Ω, Ω=B(0,1). for a class off(r, u).     

On a singular elliptic problem involving critical growth in {\mathbb{R}^{\bf N}}

In this paper, we prove a suitable Trudinger–Moser inequality with a singular weight in \mathbbR^N{\mathbb{R}^N} and as an application of this result, using the mountain-pass theorem we establish sufficient conditions for the existence of nontrivial solutions to quasilinear elliptic partial differential equations of the form

非线性四阶周期边值问题的最优正解

该文使用锥不动点定理研究了四阶周期边值问题u⁽⁴)-m⁴u+F(t, u(τ(t)))=0, 0 < t < 2π, u⁽ⁱ⁾(0)=u⁽ⁱ⁾(2π),~ i=0,1, 2, 3, 这里 F: [0,2π ]×R⁺→R⁺ 和τ: [0, 2π]→[0, 2π] 是连续的, 0-7.     

Simultaneous Representation and Approximation Formulas and High-Order Sobolev Embedding Theorems on Stratified Groups

We give various integral representation formulas simultaneously for a function and its derivatives in terms of vector field gradients of the function of appropriately high order. When the function has compact support, simpler formulas can be derived. Many of the results proved here appear to be new even in the special case of classical Euclidean space. For instance, Theorem 2.2 below reduces to the following result in the usual Euclidean case: Let B be a ball in R^N with radius r(B), let m be a positive integer, and let f\in W^m,1(B). Then there is a polynomial P=_m(B,f) of degree m-1 such that for any integers i, j with 0 j < i m and a.e. x\in B, |\nabla^j(f-P)(x)| C\int_{B}\frac{|\nablaⁱf(y)|}{|x-y|^N-(i-j)} dy +Cr(B)^i-j-N\int_B|\nablaⁱf(y)| dy. Moreover, if 0 < i-j N, then for a.e. x \in B we have the more refined formula |\nabla^j(f-P)(x)| C\int_B\frac{|\nablaⁱf(y)|}{|x-y|^N-i+j} dy.