Multiple positive solutions for a class of nonlinear four-point boundary value problem with <Emphasis Type="Italic">p</Emphasis>-Laplacian

This paper deals with the existence of multiple positive solutions for a class of nonlinear singular four-point boundary value problem with p-Laplacian：
{（φ（u′））′＋a（t）f（u（t））=0, 0〈t〈1,
αφ（u（0））-βφ（u′（ξ））=0,γφ（u（1））＋δφ（u′（η））0,
where φ（x） = ｜x｜^p-2x,p 〉 1, a（t） may be singular at t = 0 and/or t = 1. By applying Leggett-Williams fixed point theorem and Schauder fixed point theorem, the sufficient conditions for the existence of multiple （at least three） positive solutions to the above four-point boundary value problem are provided. An example to illustrate the importance of the results obtained is also given.     

一类三阶常微分方程非线性三点边值问题解的存在性

In this paper, existence of solutions of third-order differential equation
y′″（t）=f（t,y（t）,y′（t）,y″（t））
with nonlinear three-point boundary condition
{g（y（a）,y′（a）,y″（a））=0,
h（y（b）,y′（b））=0,
I（y（c）,y′（c）,y″（c））=0
is obtained by embedding Leray-Schauder degree theory in upper and lower solutions method,where a, b, c∈ R,a〈 b〈 c; f ： [a,c]×R^3→R,g：R^3→R,h：R^2→R and I：R^3→R are continuous functions. The existence result is obtained by defining the suitable upper and lower solutions and introducing an appropriate auxiliary boundary value problem. As an application, an example with an explicit solution is given to demonstrate the validity of the results in this paper.     

Blow-Up of Solutions for a Singular Nonlocal Viscoelastic Equation

We study the nonlinear one-dimensional viscoelastic nonlocal problem： uff-1/x（xux）x＋∫t0g（t-s）1/x）xux（x,s））xds=｜u｜p-2u,
with a nonlocal boundary condition. By the method given in [1, 2], we prove that there are solutions, under some conditions on the initial data, which blow up in finite time with nonpositive initial energy as well as positive initial energy. Estimates of the lifespan of blow-up solutions are also given. We improve a nonexistence result in Mesloub and Messaoudi [3].     

弱半正三阶三点边值问题的正解

The positive solutions are studied for the nonlinear third-order three-point boundary value problem u′″（t）=f（t,u（t））,a.e,t∈[0,1],u（0）=u′（η）=u″（1）=0, where the nonlinear term f（t, u） is a Caratheodory function and there exists a nonnegative function h ∈ L^1[0, 1] such that f（t, u） 〉 ≥-h（t）. The existence of n positive solutions is proved by considering the integrations of ＂height functions＂ and applying the Krasnosel＇skii fixed point theorem on cone.     

Multiple positive solutions for superlinear second order singular boundary value problems with derivative dependence

The existence of at least two positive solutions is presented for the singular second-order boundary value problem
{1/p（t）（ p（t）x′（t））′＋Φ（t）f（t,x（t）,p（t）x′（t））=0,0〈t〈1,
limt→0 p（t）x′（t）=0,x（1）=0
by using the fixed point index, where f may be singular at x = 0 and px ′= 0.     

一类多点边值问题的可解性

This paper deals with the existence of solutions for the problem
{（Фp（u′））′=f（t,u,u′）,t∈（0,1）,
u′（0）=0,u（1）=∑i=1^n-2aiu（ηi）,
where Фp（s）=｜s｜^p-2s,p〉1.0〈η1〈η2〈…〈ηn-2〈1,ai（i=1,2,…,n-2）are non-negative constants and ∑i=1^n-2ai=1.Some known results are improved under some sign and growth conditions. The proof is based on the Brouwer degree theory.     

Existence of Solutions for Schrodinger-Poisson Systems with Sign-Changing Weight

We study the existence of solutions for the SchrOdinger-Poisson system
{-△u＋u＋k（x）φu=a（x）｜u｜p-1u,in R3,
-△φ=k（x）u2, in R3,
where 3 G p 〈 5, a （x） is a sign-changing function such that both the supports of a＋ and a- may have infinite measure. We show that the problem has at least one nontrivial solution under some assumptions.     

Low Regularity Solution of the Initial-Boundary-Value Problem for the ＂Good＂ Boussinesq Equation on the Half Line

we study an initial-boundary-value problem for the ＂good＂ Boussinesq equation on the half line
{δt^2u-δx^2u＋δx^4u＋δx^2u^2=0,t〉0,x〉0.
u（0,t）=h1（t）,δx^2u（0,t） =δth2（t）,
u（x,0）=f（x）,δtu（x,0）=δxh（x）.
The existence and uniqueness of low reguality solution to the initial-boundary-value problem is proved when the initial-boundary data （f, h, h1, h2） belong to the product space
H^5（R^＋）×H^s-1（R^＋）×H^s/2＋1/4（R^＋）×H^s/2＋1/4（R^＋）
1 The analyticity of the solution mapping between the initial-boundary-data and the with 0 ≤ s 〈 1/2. solution space is also considered.     

Complexity of asymptotic behavior of solutions for the porous medium equation with absorption

In this paper we analyze the large time behavior of nonnegative solutions of the Cauchy problem of the porous medium equation with absorption u t-u m + γup = 0, where γ≥ 0, m 1 and p m + 2/N . We will show that if γ = 0 and 0 μ 2N/(N(m-1)+2), or γ 0 and 1/(p-1)μ2N/(N(m-1)+2), then for any nonnegative function φ in a nonnegative countable subset F of the Schwartz space S (R N ), there exists an initial-value u0 ∈C(RN) with lim x →∞ u 0 (x) = 0 such that φ is an ω-limit point of the rescaled solutions t μ/2 u(t β·, t), where β =[2-μ(m-1)]/4 .     

GLOBAL ASYMPTOTICS TOWARD THE REREFACTION WAVES FOR A PARABOLIC-ELLIPTIC SYSTEM RELATED TO THE CAMASSA-HOLM SHALLOW WATER EQUATION

This article is concerned with the global existence and large time behavior of solutions to the Cauchy problem for a parabolic-elliptic system related to the Camassa-Holm shallow water equation {ut＋（u^2/2）x＋px=εuxx, t〉0,x∈R, -αPxx＋P=f（u）＋α/2ux^2-1/2u^2, t〉0,x∈R, （E） with the initial data u（0,x）=u0（x）→u±, as x→±∞ （I） Here, u_ 〈 u＋ are two constants and f（u） is a sufficiently smooth function satisfying f＂ （u） 〉 0 for all u under consideration. Main aim of this article is to study the relation between solutions to the above Cauchy problem and those to the Riemann problem of the following nonlinear conservation law It is well known that if u_ 〈 u＋, the above Riemann problem admits a unique global entropy solution u^R（x/t） u^R（x/t）={u_,（f′）^-1（x/t）,u＋, x≤f′（u_）t, f′（u_）t≤x≤f′（u＋）t, x≥f′（u＋）t. Let U（t, x） be the smooth approximation of the rarefaction wave profile constructed similar to that of [21, 22, 23], we show that if u0（x） - U（0,x） ∈ H^1（R） and u_ 〈 u＋, the above Cauchy problem （E） and （I） admits a unique global classical solution u（t, x） which tends to the rarefaction wave u^R（x/t） as → ＋∞ in the maximum norm. The proof is given by an elementary energy method.     

New Monotonicity Formulae for Semi-linear Elliptic and Parabolic Systems

The authors establish a general monotonicity formula for the following elliptic system
△ui＋fi（x,ui,…,um）=0 in Ω,
where Ω belong to belong to R^n is a regular domain, （fi（x, u1,... ,um）） = △↓F（x,→↑u）, F（x,→↑u ） is a given smooth function of x ∈ R^n and →↑u = （u1,…,um） ∈ R^m. The system comes from understanding the stationary case of Ginzburg-Landau model. A new monotonicity formula is also set up for the following parabolic system
δtui-△ui-fi（x,ui,…,um）=0 in（ti,t2）×R^n,
where t1 〈 t2 are two constants, （fi（x,→↑u ） is given as above. The new monotonicity formulae are focused more attention on the monotonicity of nonlinear terms. The new point of the results is that an index β is introduced to measure the monotonicity of the nonlinear terms in the problems. The index β in the study of monotonieity formulae is useful in understanding the behavior of blow-up sequences of solutions. Another new feature is that the previous monotonicity formulae are extended to nonhomogeneous nonlinearities. As applications, the Ginzburg-Landau model and some different generalizations to the free boundary problems are studied.     

Traveling Waves and Capillarity Driven Spreading of Shear-Thinning Fluids

We study capillary spreadings of thin films of liquids of power-law rheology. These satisfy ut＋（u^λ＋2｜uxxx｜^λ-1uxxx）x=0,where u （x, t） represents the thickness of the one-dimensional liquid and λ 〉 1. We look for traveling wave solutions so that u（x,t） =g（x＋ct） and thus g satisfies g＇＇＇=｜g-ε｜^1/λ/g^1＋2/λ sgn（g-ε） We show that for each ε 〉 0 there is an infinitely oscillating solution, gε, such that limt→∞ gε=ε and that gε→ g0 as ε → O, where g0≡ 0 for t ≥ 0 and g0=cλ｜t｜3λ/2λ＋1 for t〈0 for some constant cλ.     

Blowing up of solutions to the Cauchy problem for the generalized Zakharov system with combined power-type nonlinearities

This paper deals with blowing up of solutions to the Cauchy problem for a class of general- ized Zakharov system with combined power-type nonlinearities in two and three space dimensions. On the one hand, for c0 = +∞ we obtain two finite time blow-up results of solutions to the aforementioned system. One is obtained under the condition α≥ 0 and 1 + 4/N ≤ p N +2/N-2 or α 0 and 1 p 1 + 4/N (N = 2, 3); the other is established under the condition N = 3, 1 p N +2/N-2 and α(p-3) ≥ 0. On the other hand, for c0 +∞ and α(p-3) ≥ 0, we prove a blow-up result for solutions with negative energy to the Zakharov system under study.     

奇异四阶微分方程边值问题的正解

The singular boundary value problem
{φ^（4）（x）-h（x）f（φ（x））=0,0〈x〈1,
φ（0）=φ（1）=φ′（0）=φ′（1）=0.
is considered under some conditions concerning the first eigenvaiues corresponding to the relevant linear operators, where h（x） is allowed to be singular at both x = 0 and x = 1. The existence results of positive solutions are obtained by means of the cone theory and the fixed point index.     

Regional, Single Point, and Global Blow-Up for the Fourth-Order Porous Medium Type Equation with Source

Blow-up behaviour for the fourth-order quasilinear porous medium equation with source,ut=-（｜u｜^nu）xxxx＋｜u｜^p-1u in R×R＋,where n 〉 0, p 〉 1, is studied. Countable and finite families of similarity blow-up patterns of the form us（x,t）=（T-t）^-1/p-1f（y）,where y=x/（T-t）^β＇ β=p-（n＋1）/4（p-1）,which blow-up as t → T^- 〈∞ are described. These solutions explain key features of regional （for p = n＋1）, single point （for p 〉 n＋1）, and global （for p ∈ （1,n＋1））blowup. The concepts and various variational, bifurcation, and numerical approaches for revealing the structure and multiplicities of such blow-up patterns are presented.     

BLOW-UP RATE OF SOLUTIONS FOR P-LAPLACIAN EQUATION

In this note we consider the blow-up rate of solutions for p-Laplacian equation with nonlinear source, ut = div（｜↓△u｜p-2↓△u）＋uq, （x, t） ∈ RN × （0, T）, N ≥ 1. When q 〉 p - 1, the blow-up rate of solutions is studied.     

Global-in-time Uniform Convergence for Linear Hyperbolic-Parabolic Singular Perturbations

We consider the Cauchy problem εu^″ε ＋ δu′ε ＋ Auε = 0, uε（0） = uo, u′ε（0） = ul, where ε 〉 0, δ 〉 0, H is a Hilbert space, and A is a self-adjoint linear non-negative operator on H with dense domain D（A）. We study the convergence of （uε） to the solution of the limit problem ,δu＇ ＋ Au = 0, u（0） = u0. For initial data （u0, u1） ∈ D（A1/2）× H, we prove global-in-time convergence with respect to strong topologies. Moreover, we estimate the convergence rate in the case where （u0, u1）∈ D（A3/2） ∈ D（A1/2）, and we show that this regularity requirement is sharp for our estimates. We give also an upper bound for ｜u′ε（t）｜ which does not depend on ε.     

Existence and Nonuniqueness of Weak Solutions of the Initial-Boundary Value Problem for ut=u^σ div（｜△↓u｜^（p-2）△↓u）

In this paper, we study the existence and nonuniqueness of weak solutions of the initial and boundary value problem for ut =u^σ div(|△↓u|^(p-2)△↓u) with σ≥1. Localization property of weak solutions is also discussed.     

Symmetric Positive Solutions for a Singular Second-Order Three-Point Boundary Value Problem

In this paper, we consider the following second order three-point boundary value problem u″（t）＋a（t）f（u（t））=0,0〈t〈1,u（0）-u（1）=0,u＇（0）-u＇（1）=u（1/2）,where a ： （0, 1） → [0, ∞） is symmetric on （0, 1） and may be singular at t = 0 and t = 1, f ： [0, ∞） → [O, ∞） is continuous. By using Krasnoselskii＇s fixed point theorem ia a cone, we get some existence results of positive solutions for the problem. The associated Green＇s function for the three-point boundary value problem is also given.     

Critical Exponent for the Parabolic Equation u_t=Δu~m + h(t)u~p in a Cone

In this paper,we study the initial-boundary value problem of porous medium equation ut = Δum + h(t)up in a cone D =(0,∞) ×Ω,where h(t) ～ tσ.Let ω1 denote the smallest Dirichlet eigenvalue for the Laplace-Beltrami operator on Ω and let l denote the positive root of l 2 +(n 2)l = ω1.We prove that if m p ≤ m + 2(σ+1) n+l + σ(m 1),then the problem has no global nonnegative solutions for any nonnegative u0 unless u0 = 0;if p m + 2(σ+1) n+l + σ(m1),then the problem has global solutions for some u 0 ≥ 0.