Remarks on Local Regularity for Two Space Dimensional Wave Maps

In this paper, we continue to study the equation ◻Φ^I+f^I(Φ,∂Φ) = 0 where ◻ = -∂²_t + Δ denotes the standard D' Alembertian in R^{2+1} and the nonlinear terms f have the form f^I = Σ_{JK}Γ^I_{JK}(Φ)Q_0(Φ^J,Φ^K) with Q_0(Φ,φ) = -∂_tΦ∂_tφ + Σ&sup_{i=1}∂_iΦ∂_tφ and Γ^I_{JK} being C^∞ function of Φ. In Y. Zhou [1], we showed that the initial value problem Φ(0,x) = Φ_0(x), ∂_tΦ(0,x) = Φ_1 (x) is locally well posed for Φ_0 ∈ H^{s+1}, Φ_1 ∈ H^s with s = \frac{1}{8}. Here, we shall further prove that the initial value problem is locally well posed for any s > 0.     

集值映射的不动点指数与带间断非线性项的椭圆型方程的多重解

<正> 本文研究二阶半线性椭圆边值问题■的多重解(符号详见§3),其中φ(x,t)允许对t是不连续的.一些自由边界问题可以化归这类问题.为了统一处理φ(x,t)对t连续与不连续两种情形,我们采用集值映射的观点.为此推广了经典的算子与Hammerstein算子到集值映射,并发展了集值映射的Leray-Schauder度理论;与已有的集值映射理论不同,现在处理的是映射串(定     

On the Cauchy Problem and Initial Trace for Nonlinear Filtration Type with Singularity

In this paper, we consider the Cauchy problem \frac{∂u}{∂t} = Δφ(u) in R^N × (0, T] u(x,0} = u_0(x) in R^N where φ ∈ C[0,∞) ∩ C¹(0,∞), φ(0 ) = 0 and (1 - \frac{2}{N})^+ < a ≤ \frac{φ'(s)s}{φ(s)} ≤ m for some a ∈ ((1 - \frac{2}{n})^+, 1), s > 0. The initial value u_0 (z) satisfies u_0(x) ≥ 0 and u_0(x) ∈ L¹_{loc}(R^N). We prove that, under some further conditions, there exists a weak solution u for the above problem, and moreover u ∈ C^{α, \frac{α}{2}}_{x,t_{loc}} (R^N × (0, T]) for some α > 0.     

Generalized Solution of the First Boundary Value Problem for Parabolic Monge-Ampere Equation

The existence and uniqueness of generalized solution to the first boundary value problem for parabolic Monge-Ampère equation - ut det D²_xu = f in Q = Ω × (0, T], u = φ on ∂_pQ are proved if there exists a strict generalized supersolution u_φ, where Ω ⊂ R^n is a bounded convex set, f is a nonnegative bounded measurable function defined on Q, φ ∈ C(∂_pQ), φ(x, 0) is a convex function in \overline{\Omega}, ∀x_0 ∈ ∂Ω, φ(x_0, t) ∈ C^α([0, T]).     

带有非紧条件的拟线性Schrodinger-Poisson系统非平凡解的存在性

本文研究了如下带有非紧条件的拟线性Schrodinger-Poisson系统{-△u+V(x)u+Фu+k/2u△u2=λ|u|^p-2u+f(u),x ∈R^3,-ΔФ=u^2,x∈R^3, 其中κ<0,λ>0,p≥12,f∈C(R,R),V∈C(R3,R).文中首先构造截断函数,利用集中紧性原理和逼近的方法,得到了截断后系统非平凡解的存在性;然后利用Moser迭代技巧,讨论上述系统非平凡解的存在性.     

Global W2,p Solutions of GBBM Equations on Unbounded Domain

In this paper we study the initial boundary value problem of GBBM equations on unbounded domain u_t - Δu_t = div f(u) u(x,0) = u_0(x) u|_{∂Ω} = 0 and corresponding Cauchy problem. Under the conditions: f( s) ∈ C^sup1 and satisfies (H)\qquad |f'(s)| ≤ C|s|^ϒ, 0 ≤ ϒ ≤ \frac{2}{n-2} if n ≥ 3; 0 ≤ ϒ < ∞ if n = 2 u_0(x) ∈ W^{2,p}(Ω) ∩ W^{2,2}(Ω) ∩ W^{1,p}_0(Ω)(W^{2,p}(R^n) ∩ W^{2,2}(R^n) for Cauchy problem), 2 ≤ p < ∞, we obtain the existence and uniqueness of global solution u(x, t) ∈ W^{1,∞}(0, T; W^{2,p}(Ω) ∩ W^{2,2}(Ω) ∩ W^{1,p}_0(Ω))(W^{1,∞}(0, T; W^{2,p}(R^n) ∩ W^{2,2} (R^n)) for Cauchy problem), so the results of [1] and [2] are generalized and improved in essential.     

非线性扩散方程静态解的稳定性

在本文中我们考虑下列非线性扩散方程在时间充分长时的性态u_t=(φ(u))_xx+φ(u),(x∈R,t∈R+=(0,+∞))其中函数φ(u)和φ(u)允许此方程具有行波解.首先我们给出该方程柯西问题的广义解的存在性、唯一性和一些比较原理.然后给定φ(u)的某些条件,我们证明了一些阀值效应.由这些结果我们可以看到在这些假设条件下,静态解u=a稳定的,而u=0或u=1是不稳定的,等等.     

Exponential Attractor for Complex Ginzburg-Landau Equation in Three-dimensions

In this paper, we consider the complex Ginzburg-Landau equation (CGL) in three spatial dimensions u_t = ρu + (1 + iϒ )
Δu - (1 + iμ) |u|^{2σ} u, \qquad(1) u(0, x) = u_0(x), \qquad(2) where u is an unknown complex-value function defined in 3+1 dimensional space-time R^{3+1}, Δ is a Laplacian in R³, ρ > 0, ϒ, μ are real parameters. Ω ∈ R³ is a bounded domain. We show that the semigroup S(t) associated with the problem (1), (2) satisfies Lipschitz continuity and the squeezing property for the initial-value problem (1), (2) with the initial-value condition belonging to H²(Ω ), therefore we obtain the existence of exponential attractor.     

一类非线性m-点边值问题正解的存在性

设α∈C[0,1],b∈C([0,1],(-∞,0)).设φ(t)为线性边值问题 u″+a(t)u′+b(t)u=0, u′(0)=0,u(1)=1的唯一正解.本文研究非线性二阶常微分方程m-点边值问题 u″+a(t)u′+b(t)u+h(t)f(u)=0, u′(0)=0,u(1)-sum from i=1 to(m-2)((a_i)u(ξ_i))=0正解的存在性.其中ξ_i∈(0,1),a_i∈(0,∞)为满足∑_(i=1)~(m-2)a_iφ_1(ξ_i)<1的常数,i∈{1,…,m-2}.通过运用锥上的不动点定理,在f超线性增长或次线性增长的前提下证明了正解的存在性结果.     

Lp-Lq Estimates for a Linear Perturbed Klein-Gordon Equation

We consider L^p-L^q estimates for the solution u(t,x) to tbe following perturbed Klein-Gordon equation ∂_{tt}u - Δu + u + V(x)u = 0 \qquad x∈ R^n, n ≥ 3 u(x,0) = 0, ∂_tu(x,0) = f(x) We assume that the potential V(x) and the initial data f(x) are compact, and V(x) is sufficiently small, then the solution u(t,x) of the above problem satisfies ||u(t)||_q ≤ Ct^{-a}||f||_p for t > 1 where a is the piecewise-linear function of 1/p and 1/q.     

Regularity Results for a Strongly Degenerate Parabolic Equation

M. Bertsch & R. Dal Passo proved the existence and uniqueness of the Cauchy problem for u_t = (φ(u),ψ(u_x))_x, where φ > 0, ψ is a strictly increasing function with lim_{s → ∞}ψ(s) = ψ_∞ < ∞. The regularity of the solution has been obtained under the condition φ" < 0 or φ = const. In the present paper, under the condition φ" ≤ 0, we give some regularity results. We show that the solution can be classical after a finite time. Further, under the condition φ" ≤ -α_0 (where -α_0 is a constant), we prove the gradient of the solution converges to zero uniformly with respect to x as t → +∞.     

Kirchhoff方程单峰解的局部唯一性

该文主要证明了以下非线Kirchhoff问题的单峰解的局部唯一性-(∈²a+∈b∫_R³|▽u|²dx)△u+u=K(x)|u|^p-1u,u> 0,x∈R³,其中∈> 0任意小,a,b> 0,1

3→R是连续有界函数.该文主要采用反证法结合局部的Pohozeav恒等式进行证明.     

Convergence of Iterative Difference Method with Nonuniform Meshes for Quasilinear Parabolic Systems

In this paper, we study the general difference schemes with nonuniform meshes for the following problem: u_t = A(x,t,u,u_x)u_{xx}, + f(x,t,u,u_x), 0 < x < l, 0 < t ≤ T \qquad (1) u(0,t) = u(l ,t) = 0, 0 < t ≤ T \qquad\qquad (2) u(x,0) = φ(x), 0 ≤ x ≤ l \qquad\qquad (3) where u, φ, and f are m-dimensional vector valued functions, u_t = \frac{∂u}{∂t}, u_x = \frac{∂u}{∂x}, u_{xx} = \frac{∂²u}{∂_x²}. In the practical computation, we usually use the method of iteration to calculate the approximate solutions for the nonlinear difference schemes. Here the estimates of the iterative sequence constructed from the iterative difference schemes for the problem (1)-(3) is proved. Moreover, when the coefficient matrix A = A(x, t, u) is independent of u_x, t he convergence of the approximate difference solution for the iterative difference schemes to the unique solution of the problem (1)-(3) is proved without imposing the assumption of heuristic character concerning the existence of the unique smooth solution for the original problem (1)-(3).     

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.     

REGULARIZATION OF AN ILL-POSED HYPERBOLIC PROBLEM AND THE UNIQUENESS OF THE SOLUTION BY A WAVELET GALERKIN METHOD

We consider the problem K(x)u xx = u tt , 0 < x < 1, t ≥ 0, with the boundary condition u(0,t) = g(t) ∈ L 2 (R) and u x (0, t ) = 0, where K(x) is continuous and 0 < α≤ K (x) < +∞. This is an ill-posed problem in the sense that, if the solution exists, it does not depend continuously on g. Considering the existence of a solution u(x, ) ∈ H 2 (R) and using a wavelet Galerkin method with Meyer multiresolution analysis, we regularize the ill-posedness of the problem. Furthermore we prove the uniqueness of the solution for this problem.     

一类拟线性退化抛物方程的Cauchy问题

对方程α（xx）u＋uαyu-αtu=f（·,u）,（x,y,t）∈R^2×（0,T）的Cauchy问题,给出了一个整体BV-弱解的定义,证明了该弱解的连续性.     

MULTIPLE SOLUTIONS FOR THE SCHR?DINGER-POISSON EQUATION WITH A GENERAL NONLINEARITY

We are concerned with the nonlinear Schrodinger-Poisson equation{-△u+(V(x)-λ)u+φ(x)u = f(u),(P)-△ φ = u2,limx|→+∞ φ(x)= 0,x∈ R3,where λ is a parameter,V(x)is an...     

Positive Solution of a Semilinear Elliptic Equation on RN

In this paper, we obtain the existence of positive solution of {-Δu = b(x)(u - λ)^p_+,\qquad x ∈ R^N λ > 0, |∇ u| ∈ L² (R^N),\qquad u ∈ L\frac{2N}{N-2} (R^N) under the assumptions that 1 < p < \frac{N+2}{N-2}, N ≥ 3, b(x) satisfies b(x) ∈ C(R^N), b(x) > 0 in R^N b(x) →_{|x|→∞}b^∞ and b(x) > \frac{4}{p+3}b^∞ for x ∈ R^N     

一类二阶多点时标边值问题无界解的存在性

借助不动点定理研究边值问题(φp(u△(t)))▽+f(t,u(t))=0,t∈(0,∞)Tu(0)=∑m-2i=1αiu(ηi),φp(u△(∞))=∑m-2i=1βiφp(u△(ηi))多个正解的存在性,得到了正解存在的充分条件.     

一类非线性Schrodinger-Maxwell方程组半经典解的存在性

本文将研究如下非线性Schrodinger—Maxwell方程组问题 {-ε^2△u＋V（x）u＋K（x）Фu=｜u｜^p-2u, x∈R^3, -△Ф=4πK（x）u^2, x∈R^3. 当势函数V（x）和电量函数K（x）满足一定假设条件时,作者利用变分法证明了ε充分小时,该方程组半经典解的存在性．