一类奇异半线性反应扩散方程初值问题整体解的存在唯一性及解的增长性

本文讨论如下问题其中σ＞０,０＜ｐ＜１;ｆ（ｘ）非负且ｆ（ｘ）∈∞（ＲＮ）,得到了整体解的存在唯一性及解的增长性．     

一类半线性热方程整体解的存在性与非存在性

本文研究半线性热方程的初值问题u_t－△u＝u~γ＋cu,（γ＞1）;u（x,0）=（x）非负整体L~P解的存在性与非存在性．首先证明,若C＞0,则不存在非负整体解．而后,对C＜0情形给出了解的整体存在与非存在的充分条件,特别证明了,若P＞（γ一1）或,则当。充分小时存在非负整体L~P解.最后,对系数C和初值（x）得到无穷多个门槛结果．     

半线性热方程整体解的存在性与非存在性

本文研究半线性热方程的初值问题ｕｔ－Δｕ＝ｕγ（γ＞１），ｕ（ｘ，０）＝φ（ｘ）的非负古典解与Ｌｐ解的整体存在性、非存在性与ｂｌｏｗ－ｕｐ．首先，利用归一化的高斯函数得到了一些新的解的非整体存在的充分条件，这些条件对古典解与Ｌｐ解是普遍适用的；然后又讨论了某些非负整体解的存在性．本文不但得到了一些新的结果，而且还简化和统一了某些已知结果的证明．     

一类非自治迭代泛函微分方程

本文研究了一类非自治迭代泛函微分方程ｘ′（ｔ）＝（ｘ２（ｔ）－ｔ２）ｆ（ｘ（ｘ（ｔ）））（其中ｆ∈Ｃ（Ｒ，Ｒ），单调递增，ｚｆ（ｚ）＞０，ｚ≠０）满足初始条件ｘ（σ）＝σ＞０，或满足初始条件ｘ（σ）＝－σ＜０的解的性态、解的存在性及延拓问题．     

非线性波动与神经传播混合型方程的整体紧吸引子

本文研究非线性波动与神经传播混合型方程u_tt＝u_xxt＋σ（u_x）_x－h（u）u_t-f（u）＋g（x）初边值问题的整体吸引子.在σ∈C~2,σ＇（s）＞σo＞0及h（s）∈C~1,－Co＜）且∫~u_oh（s）sds＞0）条件下我们得到了与该方程相应的动力系统整体紧吸引子的存在性，并证明了它具有有限的Hausdorff维数和fractal维数．     

一类拟线性退化抛物型方程的初边值问题

本文讨论下面方程的初边值问题。光滑有界区域，且ｐ＞２．本文用正则化方法证明了当ｐ＞α＋２时广义解的全局存在性，对ｐ＜α＋２的情形，证明当初值ｕ＿ｏ（ｘ）属于一稳定集Ｗ时广义解的全局存在性，而当ｕ＿ｏ（ｘ）充分大时广义解只局部存在并在有限时间内爆破。最后用共轭算子法得到了广义解的唯一性定理。     

无周期解的Liénard系统

本文讨论了Ｌｉｎａｒｄ方程¨ｘ＋ｆ（ｘ）·ｘ＋ｇ（ｘ）＝０周期解的不存在性，给出了该方程在Ｆ（ｘ）＝∫ｘ０ｆ（ｕ）ｄｕ有无穷多个零点ｘｎ→０（ｎ→∞），或存在δ＞０使当０＜｜ｘ｜＜δ时Ｆ（ｘ）＞０（或＜０）这二种情况下无周期解的充分条件．     

奇异非线性常微分方程非线性边界条件非负解的存在性

本文证明了非负解的存在性．这里（S）＝|S|P－2S，P＞1．     

一类半线性抛物型方程组解的blow—up

设是En中的有界域，边界充分光滑，满足内切域条件，n是的单位外法向．研究半线性抛物型方程组的初边值问题的解在有限时间内blow－up的问题．设常数a＞0,a＞-1，β＞-1；且aβ＞1．ci（x，t）＞-0（i＝一1，2）关于x，rHlder连续，uo（x）＞0，vo（x）且uo（x），vo（x）C1（）Bu是u或Bu=且初边值满足相容条件．关于方程组解在有限时间内的blow-up问题，已有不少工作[1-3]，本文用特征函数法研究问题（1）-（4）的解在有限时间内blow-up的条件．定理1问题（1）-（4）存在唯一的非负解.证显然V（x，t）=（）=（0，0）是问题（1）-（4）的…     

一类奇异半线性反应扩散方程初值问题整体解的存在唯一性及解的增长性

本文讨论如下问题其中σ>0,0<p<1,f(x))非负且f(z)∈L∞(RN),得到了整体解的存在唯一性及解的增长性.     

具奇异系数的反应扩散方程组Cauchy问题

该文讨论如下具有奇异系数的反应扩散方程组Cauchy问题非负局部解的存在性和不存 在性, 以及解在有限时间内的爆破问题(u_t-t^{-1}Δ u=α_1u^{q_1}+β_1v^\{p_1}+f_1(x),t>0,x∈R^N； v_t-t^\{-1}Δ v=α_2u^\{q_2}+β_2v^{p_2}+f_2(x),t>0,x∈R^ N；lim_{t→0+}u(t,x)=lim_{t→0+}v(t,x)=0,x∈R^N. 其中p_i>1, q_i>1 (i=1, 2) , α_1≥0, α_2>0, β_1>0, β_2≥0, f_ i(x) (i=1, 2)为连续非负有界函数, (f_1(x), f_2(x))(0, 0) . 文章给出了非负局部解存在的显式条件和非负局部解不存在的比较结果, 也得到解在有限时间爆破的一些结果.     

一个具有非线性边界条件的抛物系统整体存在性和爆破

In this paper the nonnegative classical solutions of a parabolic system with nonlinear boundary conditions are discussed. The existence and uniqueness of a nonnegative classical solution are proved. And some sufficient conditions to ensure the global existence and nonexistence of nonnegative classical solution to this problem are given.     

Eigenvalue problem for finite difference equations with p-Laplacian

In this paper, we consider a discrete four-point boundary value problem $$\triangle\bigl(\phi_p\bigl(\triangle u(k-1)\bigr)\bigr)+ \lambda e(k)f\bigl(u(k)\bigr)=0,\quad k\in N(1,T),$$ subject to boundary conditions $$\triangle u(0)-\alpha u(l_{1})=0,\qquad\triangle u(T)+\beta u(l_{2})=0,$$ by a simple application of a fixed point theorem. If e(k),f(u(k)) are nonnegative, the solutions of the above problem may not be nonnegative, this is the main difficulty for us to study positive solution of this problem. In this paper, we give restrictive conditions ??l ₁??1, ??(T+1?l ₂)??1 to guarantee the solutions of this problem are nonnegative, if it has, under the conditions e(k),f(u(k)) are nonnegative. We first construct a new operator equation which is equivalent to the problem and provide sufficient conditions for the nonexistence and existence of at least one or two positive solutions. In doing so, the usual restrictions $f_{0}=\lim_{u\rightarrow 0^{+}}\frac{f(u)}{\phi_{p}(u)}$ and $f_{\infty}=\lim_{u\rightarrow\infty}\frac{f(u)}{\phi_{p}(u)}$ exist are removed.     

Blow-up of solutions of some nonlinear wave equations

We analyze the problem of blow-up of global solutions of a semilinear wave equation with a potential and with a possible degeneration at infinity for nonnegative initial data with compact support. By using the nonlinear capacity method, we prove the theorem on the nonexistence of such a solution for a subcritical and the critical nonlinearity exponent.     

关于一类具阻尼项的非线性波方程的Cauchy问题

The paper concerns with the existence, uniqueness and nonexistence of global solution to the Cauchy problem for a class of nonlinear wave equations with damping term. It proves that under suitable assumptions on nonlinear the function and initial data the above-mentioned problem admits a unique global solution by Fourier transform method. The sufficient conditions of nonexistence of the global solution to the above-mentioned problem are given by the concavity method.     

Critical Exponents for the Heat-Conduction Equation with Nonlocal,Nonlinear Perturbations

We study the conditions for the existence and nonexistence of global in time (t > 0), nonnegative solutions of the problem

Positive solutions to a second order three-point boundary value problem

The existence, nonexistence, and multiplicity of nonnegative solutions are established for the three-point boundary value problem

     

On Global Existence of Solutions of Initial Boundary Value Problem for a System of Semilinear Parabolic Equations with Nonlinear Nonlocal Neumann Boundary Conditions

We establish conditions for the existence and nonexistence of global solutions of initial boundary value problem for a system of semilinear parabolic equations with nonlinear nonlocal Neumann boundary conditions. We show that these conditions are determined by the behavior of the problem coefficients as t→∞.     

非局部的退化抛物型方程组的解的爆破和整体存在性

该文采用弱上下解方法以及正则化的技巧,研究了一类非局部的退化的抛物型方程组的解的爆破和整体存在性,给出了方程组的解的爆破指标p_c=(p₁+p₂)(q₁+q₂)-mn,证得当p_c<0时,对任意的初值,方程组的解整体存在;当p_c>0时,对充分大的初值,解在有限时刻爆破,对充分小的初值,解整体存在;当p_c=0时,若区域充分小,则方程组存在非负整体解,若区域包含了一个充分大的球, 则解在有限时刻爆破.