On a Class of Quasilinear Parabolic Equations of Second Order with Double-degeneracy

In this paper we study the first boundary value problem for nonlinear diffusion equation \frac{∂u}{∂t} + \frac{∂}{∂x}f(u) = \frac{∂}{∂x}A(\frac{∂}{∂x}B(u)) whereA(s) = ∫¹_0a(σ)dσ, B(s) = ∫¹_0b(σ)dσ with a(s) ≥ 0, b(s) ≥ 0. We prove the existence of BV solutions under the much general structural conditions lim_{s → + ∞} A(s) = +∞, lim_{s → - ∞} A(s) = -∞ Moreover, we show the uniqueness without any structural conditions.     

Travelling Wave Front Solutions for Reaction-diffusion Systems

In this paper by using upper-lower solution method, under appropriate assumptions on f and g the existence of travelling wave front solutions for the following reaction-diffusion system is proved: {u_t - u_{xx}, = f(u,v) v_t - v_{xx} = g(u, v) As an application, the necessary and sufficient condition of the existence of monotone solutions for the boundary value problem {u" + cu' + u(1 - u- rv) = 0 v" + cv' - buv = 0 u(-∞) = v(+∞) = 0 u(+∞) = v(-∞) = 1 where 0 < r < 1, 0 < b < \frac{1 - r}{r} are known constants and c is unknown constant to be obtained.     

The Jump Conditions for Second Order Quasilinear Degenerate Parabolic Equations

ln this paper we consider the model problem for a second order quasilinear degenerate parabolic equation {D_xG(u) = t^{2N-1}D²_xK(u) + t^{N-1}D_x,F(u) \quad for \quad x ∈ R,t > 0 u(x,0) = A \quad for \quad x < 0, u(x,0) = B \quad for \quad x > 0 where A < B, and N > O are given constants; K(u) =^{def} ∫^u_Ak(s)ds, G(u)=^{def} ∫^u_Ag(s)ds, and F(u) =^{def} ∫^u_Af(s)ds are real-valued absolutely continuous functions defined on [A, B] such that K(u) is increasing, G(u) strictly increasing, and \frac{F(B)}{G(B)}G(u) - F(u) nonnegative on [A, B]. We show that the model problem has a unique discontinuous solution u_0 (x, t) when k(s) possesses at least one interval of degeneracy in [A, B] and that on each curve of discontinuity, x = z_j(t) =^{def} s_jt^N, where s_j= const., j=l,2, …, u_0(x, t) must satisfy the following jump conditions, 1°. u_0(z_j(t) - 0, t) = a_j, u_0 (z_j(t) + 0, t) = b_j, and u_0(z_j(t) - 0, t) = [a_j, b_j] where {[a_j, b_j]; j = 1, 2, …} is the collection of all intervals of degeneracy possessed by k (s) in [A, B], that is, k(s) = 0 a. e. on [a_j, b_j], j = 1, 2, …, and k(s) > 0 a. e. in [A, B] \U_j[a_j, b_j], and 2°. (z_j(t)G(u_0(x, t)) + t^{2N-1}D_xK(u_0(x, t)) + t^{N-1}F(u_0(x, t)))|\frac{s=s_j+0}{s=s_j-0} = 0     

Some Problems of Nonlinear Schrodinger Equations with the Effect of Dissipation

We first consider the initial value problem of nonlinear Schrödinger equation with the effect of dissipation, and prove the existence of global generalized solution and smooth solution as some conditions respectively. Secondly, we disscuss the asymptotic behavior of solution of mixed problem in bounded domain for above equation. Thirdly, we find the “blow up” phenomenon of the solution of mixed problem for equation iu_t = Δu + βf(|u|²)u - i\frac{ϒ(t)}{2}u, \quad x ∈ Ω ⊂ R³, t > 0 i. e. there exists T_0 > 0 such that lim^{t→Γ_0} || ∇u || ²_{L_t(Ω)} = ∞. The main means are a prior estimates on fractional degree Sobolev space, related properties of operator's semigroup and some integral identities.     

On Nonlocal Boundary Value Problems for Nonlinear Integro-differential Equations of Arbitrary Fractional Order

In this paper, we prove the existence of solutions of a nonlocal boundary value problem for nonlinear integro-differential equations of fractional order given by $$ \begin{array}{ll} ^cD^qx(t) = f(t,x(t),(\phi x)(t),(\psi x)(t)), \quad 0 < t < 1,\\x(0) = \beta x(\eta), x'(0) =0, x''(0) =0, \ldots, x^{(m-2)}(0) =0, x(1)= \alpha x(\eta), \end{array}$$ where $${q \in (m-1, m], m \in \mathbb{N}, m \ge 2}$, $0< \eta <1$$ , and ${\phi x}$ and ${\psi x}$ are integral operators. The existence results are established by means of the contraction mapping principle and Krasnoselskii’s fixed point theorem. An illustrative example is also presented.     

A NECESSARY AND SUFFICIENT CONDITION FOR THE OSCILLATION OF HIGHER-ORDER NEUTRAL EQUATIONS WITH SEVERAL DELAYS

Consider the higher-order neutral delay differential equationd~t/dt~n(x(t)+sum from i=1 to lp_ix(t-τ_i)-sum from j=1 to mr_jx(t-ρ_j))+sum from k=1 to Nq_kx(t-u_k)=0,(A)where the coefficients and the delays are nonnegative constants with n≥2 even. Then anecessary and sufficient condition for the oscillation of (A) is that the characteristicequationλ~n+λ~nsum from i=1 to lp_ie~(-λτ_i-λ~n)sum from j=1 to mr_je~(-λρ_j)+sum from k=1 to Nq_ke~(-λρ_k)=0has no real roots.     

Positive solutions of nonlinear multi-point boundary value problems

This paper deals with the existence of positive solutions of nonlinear differential equation

$$\begin{aligned} u^{\prime \prime }(t)+ a(t) f(u(t) )=0,\quad 0<t <1, \end{aligned}$$

subject to the boundary conditions

$$\begin{aligned} u(0)=\sum _{i=1}^{m-2} a_i u (\xi _i) ,\quad u^{\prime } (1) = \sum _{i=1}^{m-2} b_i u^{\prime } (\xi _i), \end{aligned}$$

where \( \xi _i \in (0,1) \) with \( 0< \xi _1<\xi _2< \cdots<\xi _{m-2} < 1,\) and \(a_i,b_i \) satisfy   \(a_i,b_i\in [0,\infty ),~~ 0< \sum _{i=1}^{m-2} a_i <1,\) and \( \sum _{i=1}^{m-2} b_i <1. \) By using Schauder’s fixed point theorem, we show that it has at least one positive solution if f is nonnegative and continuous. Positive solutions of the above boundary value problem satisfy the Harnack inequality

$$\begin{aligned} \displaystyle \inf _{0 \le t \le 1} u(t) \ge \gamma \Vert u\Vert _\infty . \end{aligned}$$

     

一类带p-Laplacian的Sturm-Liouville型2m点边值问题三个正解的存在性

该文考虑了下面的具一维$p$\,-Laplacian算子的多点边值问题 $ \left\{ \begin{array}{rl} &;\disp (\phi_{p}(x'(t)))'+h(t)f(t,x(t),x'(t))=0,\hspace{3mm}01,~\alpha_{i}>0,~\beta_{i}>0,~0<\sum\limits_{i=1}^{m-1}\alpha_{i}\xi_{i}\leq1,~ 0<\sum\limits_{i=1}^{m-1}\beta_{i}(1-\eta_{i})\leq1,~0=\xi_{0} <\xi_{1}<\xi_{2}<\cdots<\xi_{m-1}<\eta_{1}<\eta_{2}<\cdots<\eta_{m-1}<\eta_{m}=1,~i=1,2,\cdots,m-1.$ 通过运用锥上的不动点定理, 该文得到了至少三个正解的存在性. 有趣的是文中的边界条件是一个新型的Sturm-Liouville型边界条件, 这类边值问题到目前为止还很少被研究.     

Studies on a multi-point boundary value problem for second order differential equation with <Emphasis Type="Italic">p</Emphasis>-Laplacian

The existence of at least one solution of the following multi-point boundary value problem

Solvability of a higher-order multi-point boundary value problem at resonance

Based on the coincidence degree theory of Mawhin, we get a new general existence result for the following higher-order multi-point boundary value problem at resonance

Solutions to m-point boundary value problems of third order ordinary differential equations at resonance

In this paper, we study the third order ordinary differential equation: $$x'''(t) = f(t,x(t),x'(t),x''(t)), t \in (0,1)$$ subject to the boundary value conditions: $$x'(0) = x'(\xi ), x'(1) = \sum\limits_{i - 1}^{m - 3} {\beta _i x'(\eta _i )} , x''(1) = 0.$$ Hereβ _i ∈R, $\sum\limits_{i = 1}^{m - 3} {\beta _i = 1, 0< \eta _1< \eta _2< \ldots< \eta _{m - 3}< 1, 0< \xi< 1} $ . This is the case dimKerL=2. When theβ _i have different signs, we prove some existence results for the m-point boundary value problem at resonance by use of the coincidence degree theory of Mawhin [12, 13]. Since all the existence results obtained in previous papers are for the case dimKerL=1, our work is new.     

On the Existence of Positive Solutions of Quasilinear Elliptic Equations with Mixed Boundary Conditions

In this paper, the existence of positive solutions for the mixed boundary problem of quasilinear elliptic equation {-div (|∇u|^{p-2}∇u) = |u|^{p^∗-2}u + f(x, u), \quad u > 0, \quad x ∈ Ω u|_Γ_0 = 0, \frac{∂u}{∂\overrightarrow{n}}|_Γ_1 = 0 is obtained, where Ω is a bounded smooth domain in R^N, ∂Ω = \overrightarrow{Γ}_0 ∪ \overrightarrow{Γ}_1, 2 ≤ p < N, p^∗ = \frac{Np}{N-p}, Γ_0 and Γ_1 are disjoint open subsets of ∂Ω.     

Multiple positive solutions for semi-positone <Emphasis Type="Italic">m</Emphasis>-point boundary value problems

In this article, we establish the existence of at least two positive solutions for the semi-positone m-point boundary value problem with a parameter u (t) + λf (t, u) = 0, t ∈ (0, 1), u (0) = sum (biu (ξ i )) from i=1 to m-2, u(1)= sum (aiu(ξ i )) from i=1 to m-2, where λ > 0 is a parameter, 0 < ξ 1 < ξ 2 < ··· < ξ m 2 < 1 with 0 相似文献   

具$p$-Laplacian 算子的多点边值问题迭代解的存在性

利用单调迭代技巧和推广的Mawhin定理得到下述带有p-Laplacian算子的多点边值问题迭代解的存在性,{(Фp(u'))' f(t,u, Tu)=0, 0(≤)t(≤)1,u(0)=q-1∑i=1γiu(δi),u(1)=m-1∑i=1ηiu(ξi),其中Фp(s)=|s|p-2s,p＞1;0＜δi＜1,γi＞0,1(≤)i(≤)q-1;0＜ξi＜1,ηi(≥)0,1(≤)i(≤)m-1且q-1∑i=1γi＜1,m-1∑i=1ηi(≤)1;Tu(t)=∫t0k(t,s)u(s)ds,k(t,s)∈C(I×I,R ).     

Identities for Bernoulli polynomials and Bernoulli numbers

We prove that if m and \({\nu}\) are integers with \({0 \leq \nu \leq m}\) and x is a real number, then

1. $$\sum_{k=0 \atop k+m \, \, odd}^{m-1} {m \choose k}{k+m \choose \nu} B_{k+m-\nu}(x) = \frac{1}{2} \sum_{j=0}^m (-1)^{j+m} {m \choose j}{j+m-1 \choose \nu} (j+m) x^{j+m-\nu-1},$$ where B _n (x) denotes the Bernoulli polynomial of degree n. An application of (1) leads to new identities for Bernoulli numbers B _n . Among others, we obtain
2. $$\sum_{k=0 \atop k+m \, \, odd}^{m -1} {m \choose k}{k+m \choose \nu} {k+m-\nu \choose j}B_{k+m-\nu-j} =0 \quad{(0 \leq j \leq m-2-\nu)}. $$ This formula extends two results obtained by Kaneko and Chen-Sun, who proved (2) for the special cases j = 1, \({\nu=0}\) and j = 3, \({\nu=0}\) , respectively.

     

BOUNDARY VALUE PROBLEMS OF SINGULARLY PERTURBED INTEGRO－DIFFERENTIAL EQUATIONS

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On Initial-boundary-value Problems for a Class of Systems of Quasi-linear Evolution Equations

In this paper the initial-boundary-value problems for pseudo-hyperbolic system of quasi-linear equations: {(-1)^Mu_{tt} + A(x, t, U, V)u_x^{2M}_{tt} = B(x, t, U, V)u_x^{2M}_{t} + C(x, t, U, V)u_x^{2M} + f(x, t, U, V) u_x^k(0,t) = ψ_{0k}(t), \quad u_x^k(l,t) = ψ_{lk}(t), \quad k = 0,1,…,M - 1 -u(x,0) = φ_0(x), \quad u_t(x,0) = φ_1(x) is studied, where U = (u_1, u_x,…,u_x^{2M - 1}) V = (u_t, u_{xt},…,u_x^{2M - 1_t}), A, B, C are m × m matrices, u, f, ψ_{0k}, ψ_{1k}, ψ_0, ψ_1 are m-dimensional vector functions. The existence and uniqueness of the generalized solution (in H² (0, T; H^{2M} (0, 1))) of the problems are proved.     

The Boundedness for Generalized Solutions of Quasilinear Elliptic Equations

Under the appropriate conditions on u, the generalized solution of the elliptic equation ∫_G {∇v ⋅ A(x, u, ∇u) + vB(x, u, ∇u)}dx = 0, \quad ∀v ∈ {\WW}¹_p(G) ∩ L_∞(G) for which even the natural growth condition p(1 - 1/p^∗) < ϒ < p is permitted, the local and global boundedness of u are proved.     

The Uniqueness of Viscosity Solutions of the Second Order Fully Nonlinear Elliptic Equations

Recently R. Jensen [1] has proved the uniqueness of viscosity solutions in W^{1,∞} of second order fully nonlinear elliptic equation F (D², Du, u) = 0. He does not assume F to be convex. In this paper we extend his result [1] to the case that F can be dependent on x, i. e. prove that the viscosity solutions in W^{1,∞} of the second order fully nonlinear elliptic equation F (D²u, Du, u, x) = 0 are unlique. We do not assume F to be convex either.