一类带有退化强制项的奇异椭圆方程解的存在性

该文研究了一类具退化强制项和自然增长条件梯度项的奇异椭圆方程的Dirichlet边值问题{-div((∣▽u∣~(p-2)▽u)/((1+∣u∣)~r)+B(∣▽u∣~p)/(∣u∣~θ)=f,x∈Ω,u0,x∈Ω,u=0,x∈аΩ,其中,ΩR~N(N≥p)是一有界区域,B,γ,θ0,P1,f是某一Lebesgue空间L~m(Ω)(m≥1)中的非负函数.利用截断技术并结合选取适当的检验函数,证明了该问题非负解的存在性以及正则性等结果.该文结果表明尽管低阶梯度项是奇异的,它的存在对解的正则性有"好"的影响.     

一类退化半导体方程弱解存在性的研究

文章研究了当■(s)=sm(m>1),b(s)=s2和初值为u0,v0∈L 2(Ω)时一类退化半导体方程弱解的存在性．文章首先将原问题正则化,然后对正则化问题在L 2(Ω)空间上做出了有界估计,最后利用收敛性得到了问题的结论．     

具有阻尼的半线性波动方程解的衰减估计

研究具有阻尼的半线性波动方程的初边值问题u_(tt)-△u+βu_t=|u|~(p-1)u,x∈Ω,t>0u(x,0)=u_0(x),u_t(x,0)=u_1(x),x∈Ωu|_((?)Ω)=0,t≥0其中γ为正常数,Ω■R~n为有界域,当n≥3时,1相似文献   

磁流体力学方程组弱解的适定性

考虑由磁流体力学方程组控制的二维不可压缩流体的初边值问题,在边界光滑的有界区域中,当(u_0,B_0)∈((W~(m,p)(Ω))~2×W~(m,p)(Ω))时,利用Galerkin方法和先验估计,得到了相应的初边值问题存在唯一的弱解(u(·,t),B(·,t))∈((W~(m,p)(Ω))~2×W~(m,p)(Ω)),并证明了弱解对初值(u_0,B_0)具有连续依赖性.     

拟凸条件下变分问题解的正则性

本文研究变分问题(1)■(u:Ω)=∫_Ωf(x,u,Du)dx的极小函数的正则性,其中Ω■R~n是有界开域,u:Ω→R-~N,Du:Ω→R(nN),f: ×R~N×R(nN)→R。定义称函数f满足严格拟凸条件,是指存在常数v>0,使得对任意的(x_0,u_0,p_0)∈Ω×R~N×R(nN)和φ∈C~∞_0(Ω,R~N),都有■(2)其中|Ω|是Ω的Lebesgue测度。定理设u∈H~(1,2)(Ω,R~N)是泛函f的极小函数,即对任意的φ∈H_0~(1,2)(Ω,R~N),都有■而f(x,u,p)满足下列假设 (H1) f满足严格拟凸性,即(2)成立, (H2) f关于p的二阶导数存在,且存在常数L>0,使得■对任意的(x,u,p)∈Ω×R~N×R~(nN),都有■ (3)|f_(pp)(x,u,p)|≤L_0 (H3) 存在[0,∞]上的连续、有界、凹的函数∞(t),使得(4)■(5)■(6)■且ω(t)≤At~α,其中A,α是正常数。那么存在常数δ∈(0,1)和开集Ω_0Ω,使得|Ω-Ω_0|=0,Du∈C~6(Ω_0,R(nN))。     

一类Boussinesq方程的整体解的存在性与不存在性

主要考察Boussinesq方程v_(tt)-v_(xx)+v_(xxx)=σ(v)_(xx),x∈R的整体解的存在性和blow-up问题,当σ(v)=-β(|v|~p v),β0,p0时,通过采用构造稳定集(位势井)W={v∈H~1(R)|||v_x||~2+||v||~22(p+2)/p d}和不稳定集V={v∈H~1(R)|||v_x||~2+||v||~22(p+2)/p d}的方法,得到了W和V在上述方程的流下是不变的,并证明了如果初始能量E(0)≤d,那么当初值v_0∈(?)时,问题存在惟一整体解;当初值v_0∈V时,问题的解在有限时刻T_1∈(t_1,t_1+4φ(t_1)/pφ′(t_1))发生爆破.     

具粘性拟线性波方程的能量衰减估计

给出下列具粘性拟线性波方程初边值问题解的能量衰减估计u_(tt)(t,x)-div{σ(|▽u(t,x)|~2)▽u(t,x)}-△u(t,x)-△ut(t,x)+δ|u_t(t,x)|~(p-1)u_t(t,x)=μ|u(t,x)|~(q-1)u(t,x),x∈Ω,t∈(0,T),u(t,x)|■Ω=0,t∈(0,T),u(0,x)=u_0(x),u_t(0,x)=u_1(x),x∈Ω,其中Ω是R~N(N≥1)中具有光滑边界■Ω的区域,p≥1,q1,δ0,μ0,△表示Laplace算子,▽表示梯度算子和σ(s)是一给定的非线性函数.证明的思想是应用一已知的积分不等式,证明以上初边值问题解的能量衰减估计.     

一类椭圆型方程不适定问题的Meyer小波正则化方法

Laplace方程Cauchy问题u_(xx)(x,y)+u_(yy)(x,y)=0,0x1,y∈R,u(0,y)=g(y),y∈R,u_x(0,y)=0,y∈R是一个严重不适定问题,数据g的微小变化可以引起解的巨大误差,本文通过构造一个在频域具紧支集的小波并在尺度空问上展开数据和解,滤除高频部分,并结合Galerkin方法,建立了一种逼近准确解的正则化方法,恢复解对初值的连续依赖性,建立误差估计.     

竞赛图具有弧泛回路性的一个充分条件

关于竞赛图的弧泛迴路性问题,Alspach证明了正则竞赛图具有此性质.朱永津、田丰证明了若竞赛图 T 中任意一个弧(v,v_0)都满足条件 d~+(v_0)+d~-(v)≥p-2,这里 p 为 T 的顶点数,则当 p≥7时,T 中过任一弧存在迴路系列 C_4,C_5,…,C_p.本文提出并证明了若 T 满足以下条件:当 d~+(v)<1/2(p-1)时,在 v 的外邻集 O(v)中有一点 u,d~+(u)≥1/2(p-1);当 d~+(v_1),d~+(v_2)<1/2(p-1)时,有 u_1,u_2∈O(v_1)∪O(v_2),d~+(u_1),d~+(u_2)≥1/2(p-1),且对入次亦满足相应的条件,则当 p≥9和最小次数δ≥4时,过 T 的每一个弧存在迴路系列 c_6,c_7,…,c_p.此充分条件不要求顶点次数的正则性和几乎正则性,对 T 的不正则度 q=(?)|d~+(v)-d~-(v)|一般来说也没有限制.     

一类N维非线性波动方程的Cauchy问题

证明下列非线性波动方程的Cauchy问题v_(tt)-α△v_(tt)-Δv=g(v)-αΔg(v),x∈R~N,t>0,(1)v(x,0)=v_0(x),v_t(x,0)=v_1(x),x∈R~N(2)在空间C~2([0,∞);H~s(R~N))(s>N/2)中存在唯一整体广义解v和在空间C~2([0,∞);H~s(R~N))(s>2+N/2N)中存在唯一整体古典解v,即u∈C~2([0,∞);C_B~2(R~N)).还证明Cauchy问题(1),(2)在C~3([0,∞);W~(m,p)(R~N)∩L~∞(R~N))(m≥0,1≤p≤∞)中有唯一整体广义解v和在C~3([0,∞);W~(m,p)(R~N)∩L~∞(R~N))(m>2+N/P)中有唯一整体古典解v,即v∈C~3([0,∞);C~2(R~N)∩L~∞(R~N)).     

Scalar conservation laws with general boundary condition and continuous flux function

We introduce a notion of entropy solution for a scalar conservation law on a bounded domain with nonhomogeneous boundary condition: ut+divΦ(u)=f on Q=(0,T)×Ω, u(0,⋅)=u₀ on Ω and “u=a on some part of the boundary (0,T)×∂Ω.” Existence and uniqueness of the entropy solution is established for any Φ∈C(R;RN), u₀∈L^∞(Ω), f∈L^∞(Q), a∈L^∞((0,T)×∂Ω). In the L¹-setting, a corresponding result is proved for the more general notion of renormalised entropy solution.     

Multiple positive solutions of nonhomogeneous elliptic systems with strongly indefinite structure and critical Sobolev exponents

In this paper, we study the existence of multiple positive solutions to some Hamiltonian elliptic systems −Δv=λu+u^p+εf(x), −Δu=μv+v^q+δg(x) in Ω;u,v>0 in Ω; u=v=0 on ∂Ω, where Ω is a bounded domain in R^N (N?3); 0?f, g∈L∞(Ω); 1/(p+1)+1/(q+1)=(N−2)/N, p,q>1; λ,μ>0. Using sub- and supersolution method and based on an adaptation of the dual variational approach, we prove the existence of at least two nontrivial positive solutions for all λ,μ∈(0,λ₁) and ε,δ∈(0,δ₀), where λ₁ is the first eigenvalue of the Laplace operator −Δ with zero Dirichlet boundary conditions and δ₀ is a positive number.     

New decay estimates for linear damped wave equations and its application to nonlinear problem

We present new decay estimates of solutions for the mixed problem of the equation v_tt?v_xx+v_t=0, which has the weighted initial data [v₀,v₁]∈(H¹₀(0,∞) ∩L^1,γ(0,∞)) × (L²(0,∞)∩L^1,γ(0,∞)) (for definition of L^1,γ(0,∞), see below) satisfying γ∈[0,1]. Similar decay estimates are also derived to the Cauchy problem in ?^N for u_tt?Δu+u_t=0 with the weighted initial data. Finally, these decay estimates can be applied to the one dimensional critical exponent problem for a semilinear damped wave equation on the half line. Copyright © 2004 John Wiley & Sons, Ltd.     

Existence of three positive pseudo-symmetric solutions for a one dimensional p-Laplacian

We apply the Five Functionals Fixed Point Theorem to verify the existence of at least three positive pseudo-symmetric solutions for the three point boundary value problem, (g(u′))′+a(t)f(u)=0, u(0)=0, and u(ν)=u(1), where g(v)=|v|^p−2v, with p>1 and ν∈(0,1).     

Hamilton cycles in digraphs of unitary matrices

A set S⊆V is called a q⁺-set (q^--set, respectively) if S has at least two vertices and, for every u∈S, there exists v∈S,v≠u such that N⁺(u)∩N⁺(v)≠∅ (N^-(u)∩N^-(v)≠∅, respectively). A digraph D is called s-quadrangular if, for every q⁺-set S, we have |∪{N⁺(u)∩N⁺(v):u≠v,u,v∈S}|?|S| and, for every q^--set S, we have |∪{N^-(u)∩N^-(v):u,v∈S)}?|S|. We conjecture that every strong s-quadrangular digraph has a Hamilton cycle and provide some support for this conjecture.     

Convergence to strong nonlinear diffusion waves for solutions to p-system with damping on quadrant

In this paper, we consider the so-called p-system with linear damping on quadrant. We show that for a certain class of given large initial data (v₀(x),u₀(x)), the corresponding initial-boundary value problem admits a unique global smooth solution (v(x,t),u(x,t)) and such a solution tends time-asymptotically, at the Lp (2?p?∞) optimal decay rates, to the corresponding nonlinear diffusion wave which satisfies (1.9) provided the corresponding prescribed initial error function (V₀(x),U₀(x)) lies in (H³(R⁺)∩L¹(R⁺))×(H²(R⁺)∩L¹(R⁺)).     

Model problem for the motion of a compressible, viscous flow with the no-slip boundary condition

We consider the Navier–Stokes equations for the motion of a compressible, viscous, pressureless fluid in the domain W = \mathbbR³₊{\Omega = \mathbb{R}^3_+} with the no-slip boundary conditions. We construct a global in time, regular weak solution, provided that initial density ρ ₀ is bounded and the magnitude of the initial velocity u ₀ is suitably restricted in the norm ||?{r₀(·)}u₀(·)||_L²(W) + ||?u₀(·)||_L²(W){\|\sqrt{\rho_0(\cdot)}{\bf u}_0(\cdot)\|_{L^2(\Omega)} + \|\nabla{\bf u}_0(\cdot)\|_{L^2(\Omega)}}.     

Existence of a unique solution to a quasilinear elliptic equation

The purpose of this paper is to prove the existence of a unique classical solution u(x) to the quasilinear elliptic equation −∇⋅(a(u)∇u)+v⋅∇u=f, where u(x₀)=u₀ at x₀∈Ω and where n⋅∇u=g on the boundary ∂Ω. We prove that if the functions a, f, v, g satisfy certain conditions, then a unique classical solution u(x) exists. Applications include stationary heat/diffusion problems with convection and with a source/sink, where the value of the solution is known at a spatial location x₀∈Ω, and where n⋅∇u is known on the boundary.     

Global solutions of the fast diffusion equation with gradient absorption terms

We investigate the existence of nonnegative weak solutions to the problem ut=Δ(um)−p|∇u| in Rn×(0,∞) with ⁺(1−2/n)<m<1. It will be proved that: (i) When 1<p<2, if the initial datum u₀∈D(Rn) then there exists a solution; (ii) When 1<p<(2+mn)/(n+1), if the initial datum u₀(x) is a bounded and nonnegative measure then the solution exists; (iii) When (2+mn)/(n+1)?p<2, if the initial datum is a Dirac mass then the solution does not exist. We also study the large time behavior of the L¹-norm of solutions for 1<p?(2+mn)/(n+1), and the large time behavior of t1/β‖u(⋅,t)−Ec(⋅,t)L^∞_‖ for (2+mn)/(n+1)<p<2.     

Asymptotic behaviour of solutions to some pseudoparabolic equations

The aim of this paper is to investigate the behaviour as t→∞ of solutions to the Cauchy problem u_t−△u_t−v△u−(b,∇u)=∇⋅F(u),u(x,0)=u₀(x), where v>0 is a fixed constant, t≥0, x∈Ω, Ω is a bounded domain in Rⁿ. We will first establish an a priori estimate. Then, we establish the global existence, uniqueness and continuous dependence of the weak solution for the Sobolev-Galpern type equation with the Dirichlet boundary.