超线性非线性Sturm-Liouville边值问题的正解

利用拓扑方法讨论了一类非线性Sturm-Liouville边值问题{-u″=λf(x,u),0≤x≤1,α0u(0)+β0u′(0)=0, α1u(1)+β1u′(1)=0.研究了上述问题的正解的全局结构,在非线性项f(x,u)不满足条件f(x,u)≥0(u≥0)时获得了正解的存在性.     

超线性非线性Sturm-Liouville边值问题的正解

利用拓扑方法讨论了一类非线性Sturm-Liouville边值问题-u″=λf(x,u),0≤x≤1,α_0u(0)+β_0u′(0)=0,α_1u(1)+β_1u′(1)=0.研究了上述问题的正解的全局结构,在非线性项f(x,u)不满足条件f(x,u)≥0(u≥0)时获得了正解的存在性.     

奇异半线性反应扩散方程初值问题的一些注记

考虑下述奇异半线性反应扩散方程初值问题(()-1-t△u=ut+f(x),t＞0,x∈RN lim u(t,x)=0,x∈RN t→0=)其中r＞0,△=∑( )/( )x2i,f(x)非负且f(x)∈L∞(RN).首先利用增算子不动点定理,重新证明了IVP在(0,+∞)上至少存在一个非负解,并给出了IVP解的迭代逼近序列.其次获得了一个有关IVP(1)正解的无限增长性的结果.最后,证明了当r＞1时,去掉条件1/r-1≥n/2,IVP的正解u(t)同样会产生爆破.研究结果表明情形limut→+∞(t,x)=+∞不会出现.     

带梯度项的半线性椭圆方程正整体解的存在性（英文）

本文应用上下解方法、摄动方法等,进一步推广了早期结果并给出半线性椭圆方程-△u+p(x)|▽u|~γ=λf(x,u),u0,x∈R~N,lim_(|x|→∞)u(x)=0,正解的存在性,其中γ∈(1,2],λ0,函数p:R~N→[0,∞)和f:R~N×(0,∞)→[0,∞)均为局部H(o|¨)lder连续.     

一类奇异边值问题的正解

考虑如下二阶非线性奇异两点边值问题{-u'+f(u)-u~(-γ)=λu,0x1,u0,u(0)=u(1)=0,}其中0γ1为常数,λ0为特征值参数.f(u)满足给定的条件.利用上下解方法和Arzela-Ascoli定理讨论二阶非线性奇异两点边值问题正解的存在性和唯一性.特别地,利用适当的变换和最大值原理给出方程在特殊形式下正解的渐近行为.     

p—Laplace方程的Neumann问题的正解

本文我们讨论p-Laplace方程-sum from i=1 to n(D_i(∣Du∣~(p-2)D_iu)=u~q+f(x,u)在Neumann边界条件D_Yu=0下的正解存在性,其中10,B>0,以及t∈(p-1,n(p-1)/(n-p)),则上述问题存在一个正解。     

具有渐进非线性项的Kirchhoff型方程的正解（英文）

本文研究一类非线性Kirchhoff型方程.非线性项函数f(x, u)在无穷远处关于u是渐进线性或渐进非线性的.若位势函数V (x)和非线性项f(x, u)满足给定的条件,本文在工作空间缺乏紧性嵌入的情形下获得该方程正解的存在性.     

一个奇异典型弹性梁方程涉及第一特征值的正解

本文研究非线性四阶边值问题u(4)(t)=f(t,u(t)),0π4/16;(ii)∫10lim inf x→+0f(t,x)/x dt>π4/16并且∫10lim sup x→+∞f(t,x)/x dt<π4/16.     

Banach空间中四阶常微分方程两点边值问题的多重解

主要利用锥不动点理论研究Banach空间中四阶常微分方程两点边值问题{x^(4)(t)=f(t,u(t)),t∈(0,1) x(0)=x‘(1)=x″(0)=x′″(1)=θ的正解及多个正解的存在性并给出了应用．     

一类四阶边值问题正解的存在性

讨论了四阶常微分方程边值问题u^(4)=βu″-au=ψ(t)f(u),u(0)=u(1)=u″（0)=u″(1)=0的正解的存在性,利用锥拉伸与锥压缩不动点定理证明了,当f(u)在u=0及u=∞超线性或次线性增长时,该问题至少存在一个正解。     

BLOW-UP RATE OF POSITIVE SOLUTION OF UNIFORMLY PARABOLIC EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS

This paper deals with the blow-up of positive solutions of the uniformly pa-rabolic equations ut = Lu + a(x)f(u) subject to nonlinear Neumann boundary conditions . Under suitable assumptions on nonlinear functi-ons f, g and initial data U0(x), the blow-up of the solutions in a finite time is proved by the maximum principles. Moreover, the bounds of "blow-up time" and blow-up rate are obtained.     

一个局部波动方程组的解的全局存在性与爆破

我们研究初始值问题(e)u1/(e)t2=(e)2u1/(e)x2+‖u2(·,t)‖p, (e)2u2/(e)t2=(e)2u2/(e)x2+‖u1(·,t)‖q,-∞<x<∞,t>0,u1(x,0)=f1(x), (e)u1/(e)t(x,0)=g1(x),u2(x,0)=f2(x), (e)u2/(e)t(x,0)=g2(x),- ∞<x<∞,where‖ui(·,t)‖=∫∞-∞(4)i(x)|ui(x,t)|dx with (4)i(x)≥0 and ∫∞-∞(4)i(x)dx=1,i=1,2.然后建立解的全局存在和爆破的标准,提出爆破增长率.     

Blow-up vs. Global Finiteness for an Evolution $p$-Laplace System with Nonlinear Boundary Conditions

In this paper, the authors consider the positive solutions of the system of the evolution $p$-Laplacian equations $$\begin{cases} u_t ={\rmdiv}(| ∇u |^{p−2} ∇u) + f(u, v), & (x, t) ∈ Ω × (0, T ), & \\ v_t = {\rmdiv}(| ∇v |^{p−2} ∇v) + g(u, v), &(x, t) ∈ Ω × (0, T) \end{cases}$$with nonlinear boundary conditions $$\frac{∂u}{∂η}= h(u, v), \frac{∂v}{∂η} = s(u, v),$$and the initial data $(u_0, v_0)$, where $Ω$ is a bounded domain in$\boldsymbol{R}^n$with smooth boundary $∂Ω, p > 2$, $h(· , ·)$ and $s(· , ·)$ are positive $C^1$ functions, nondecreasing in each variable. The authors find conditions on the functions $f, g, h, s$ that prove the global existence or finite time blow-up of positive solutions for every $(u_0, v_0)$.     

Phénomène d’explosion totale pour l’équation de la chaleur avec non-linéarité du type puissance

In this Note, we are interested in the possible continuation after the blow-up time T_m of radially symmetric positive classical solutions u of the heat equation with nonlinearity f(u) = u^p, where p > 1. We say that u blows up completely after T_m if u can not be extended beyond T_m (even in the weak sense). We obtain a complete blow up criterion which relies on the asymptotic behaviour of u around the blow-up singularity x = 0.     

Finite time blow-up and global existence of weak solutions for pseudo-parabolic equation with exponential nonlinearity

This paper is concerned with the initial boundary value problem of a class of pseudo-parabolic equation $u_t - \triangle u - \triangle u_t + u = f(u)$ with an exponential nonlinearity. The eigenfunction method and the Galerkin method are used to prove the blow-up, the local existence and the global existence of weak solutions. Moreover, we also obtain other properties of weak solutions by the eigenfunction method.     

Global Existence and Blow-up for a Parabolic System with Nonlinear Boundary Conditions

This paper deals with the global existence and blow-up of positive solutions to the systems: u_t = ∇(u^∇u) + u¹ + v^a v_t = ∇(v^n∇v) + u^b + v^k in B_R × (0, T) \frac{∂u}{∂η} = u^αv^p, \frac{∂v}{∂η} = u^qv^β on S_R × (0, T) u(x, 0) = u_0(x), v(x, 0} = v_0(x) in B_R We prove that there exists a global classical positive solution if and only if l ≤ l, k ≤ 1, m + α ≤ 1, n + β ≤ 1, pq ≤ (1 - m - α)(1 - n - β),ab ≤ 1, qa ≤ (1 - n - β) and pb ≤ (1 - m - α).     

Blow-up solutions of inhomogeneous nonlinear Schrödinger equations

In this paper, we study blow-up solutions to the Cauchy problem of the inhomogeneous nonlinear Schrödinger equation $ \partial_t u = i ( f(x) \Delta u + \nabla f(x) \cdot \nabla u +k(x)|u|^2u) $ on ${\mathbb{R}}^2In this paper, we study blow-up solutions to the Cauchy problem of the inhomogeneous nonlinear Schr?dinger equation on . We present existence and non-existence results and investigate qualitative properties of the solutions when they exist. Mathematics Subject Classification (2000) 35Q55, 35G25 Dedicated respectfully to Professor Weiyue Ding on the occasion of his sixtieth birthday.     

A symmetry-breaking problem in the annulus

We study questions of degeneracy and bifurcation for radial solutions of the semilinear elliptic equation ?u(x) + f(u(x)) = 0, x isin; [math001], [math001]an annulus in Rⁿ, with homogeneous Dirichlet boundary conditions. For certain nonlinearities f(u), we prove existence of degenerate radial solutions u (for which the kernel of the linearized operator Lz = ?z + [math001](u)z, z isin; C²We study questions of degeneracy and bifurcation for radial solutions of the semilinear elliptic equation ?u(x) + f(u(x)) = 0, x isin; [math001], [math001]an annulus in Rⁿ, with homogeneous Dirichlet boundary conditions. For certain nonlinearities f(u), we prove existence of degenerate radial solutions u (for which the kernel of the linearized operator Lz = ?z + [math001](u)z, z isin; C²$⁰([math001]), is non-trivial) and existence of nonradial solutions for the semi-linear equation. These nonradial (asymmetric) solutions are obtained via a bifurcation procedure from the radial (symmetric) ones. This phenomena is called symmetry-breaking. The bifurcation results are proved by a Conley index argument     

Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation

We consider the critical nonlinear Schrödinger equation $iu_{t} = -\Delta u-|u|^{4/N}$ with initial condition u(0, x) = u₀.For u₀$\in$H¹, local existence in time of solutions on an interval [0, T) is known, and there exist finite time blow-up solutions, that is u₀ such that $\textrm{lim} _{t\uparrow T <+\infty}|\nabla u(t)|_{L^{2}}=+\infty$. This is the smallest power in the nonlinearity for which blow-up occurs, and is critical in this sense.The question we address is to control the blow-up rate from above for small (in a certain sense) blow-up solutions with negative energy. In a previous paper [MeR], we established some blow-up properties of (NLS) in the energy space which implied a control $|\nabla u(t)|_{L^{2}} \leq C \frac{|\ln(T-t)|^{N/4}}{\sqrt{T-t}}$ and removed the rate of the known explicit blow-up solutions which is $\frac{C}{T-t}$.In this paper, we prove the sharp upper bound expected from numerics as$|\nabla u(t)|_{L^{2}} \leq C \left(\frac{\ln|\ln(T-t)|}{T-t} \right)^{1/2}$by exhibiting the exact geometrical structure of dispersion for the problem.     

Evolution equations with a free boundary condition

In this paper we consider the heat flow of harmonic maps between two compact Riemannian Manifolds M and N (without boundary) with a free boundary condition. That is, the following initial boundary value problem ∂₁,u −Δu = Γ(u)(∇u, ∇u) [tT T_{u u}N, on M × [0, ∞), u(t, x) ∈ Σ, for x ∈ ∂M, t > 0, ∂u/t6n(t, x) ⊥_u T_u(t,x) Σ, for x ∈ ∂M, t > 0, u(o,x) = u_o(x), on M, where Σ is a smooth submanifold without boundary in N and n is a unit normal vector field of M along ∂M. Due to the higher nonlinearity of the boundary condition, the estimate near the boundary poses considerable difficulties, even for the case N = ℝⁿ, in which the nonlinear equation reduces to ∂_tu-Δu = 0. We proved the local existence and the uniqueness of the regular solution by a localized reflection method and the Leray-Schauder fixed point theorem. We then established the energy monotonicity formula and small energy regularity theorem for the regular solutions. These facts are used in this paper to construct various examples to show that the regular solutions may develop singularities in a finite time. A general blow-up theorem is also proven. Moreover, various a priori estimates are discussed to obtain a lower bound of the blow-up time. We also proved a global existence theorem of regular solutions under some geometrical conditions on N and Σ which are weaker than K_N <-0 and Σ is totally geodesic in N.