奇异半线性发展方程的局部Cauchy问题

本文在Ｂａｎａｃｈ空间Ｅ中讨论如下问题ｄｕｄｔ＋１ｔσＡｕ＝Ｊ（ｕ），０＜ｔＴ，ｌｉｍｔ→０＋ｕ（ｔ）＝０，其中ｕ：（０，Ｔ］Ｅ，Ａ是与ｔ无关的线性算子．（－Ａ）是Ｅ上Ｃ０半群｛Ｔ（ｔ）｝ｔ０的无穷小生成元，常数σ１，Ｊ是一个非线性映射ＥＪ→Ｅ．它满足局部Ｌｉｐｓｃｈｉｔｚ条件．我们证明了当其Ｌｉｐｓｃｈｉｔｚ常数ｌ（ｒ）满足一定条件时．问题（Ｓ）有局部解，且在某函类中解唯一．设Ｊ（ｕ）＝｜ｕ｜γ－１ｕ＋ｆ（ｘ）（γ＞１），Ｅ＝Ｌｐ，ＥＪ＝Ｌｐγ时得到了与Ｗｅｉｓｌｅｒ［２］在非奇异情形类似的结果．     

Critical Fujita Exponents for Localized Reaction Diffusion Systems

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｉｓｐａｐｅｒｄｅａｌｓｗｉｔｈｔｈｅｆｏｌｌｏｗｉｎｇｉｎｉｔｉａｌｖａｌｕｅｐｒｏｂｌｅｍｕｔ＝Δｕ＋ｖｐ１（０,ｔ）ｖｒ１（ｘ,ｔ）,ｖｔ＝Δｖ＋ｕｐ２（０,ｔ）ｕｒ２（ｘ,ｔ）,ｕ（ｘ,０）＝ｕ０（ｘ）,ｖ（ｘ,０）＝ｖ０（ｘ）,　　ｘ∈ＲＮ,ｔ＞０,ｘ∈ＲＮ,ｔ＞０,ｘ∈ＲＮ,（１）ｗｈｅｒｅＮ１,ｐｉ＞０,ｒｉ１,ｉ＝１,２,ｕ０（ｘ）０ａｎｄｖ０（ｘ）０ａｒｅｎｏｎｎｅｇａｔｉｖｅｃｏｎｔｉｎｕｏｕｓ,ａｎｄｂｏｕｎｄｅｄｆｕｎｃｔｉｏｎ…     

求f(x)的若干方法

换元法例１已知ｆ（ｓｉｎｘ－１）＝ｃｏｓ２ｘ＋２,求ｆ（ｘ）．解设ｓｉｎｘ－１＝ｔ,∴ｓｉｎｘ＝ｔ＋１（－２≤ｔ≤０）,则ｃｏｓ２ｘ＝１－ｓｉｎ２ｘ＝１－（ｔ＋１）２,∴ｆ（ｔ）＝１－（ｔ＋１）２＋２（－２≤ｔ≤０）,∴ｆ（ｘ）＝－ｘ２－２ｘ＋２（－...     

On the Decay of Solution of Some Nonlinear Hyperbolic Equation

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎａｎｄＲｅｓｕｌｔＩｎｔｈｉｓａｒｔｉｃｌｅｗｅａｒｅｃｏｎｃｅｒｎｅｄｗｉｔｈｔｈｅｄｅｃａｙｏｆｇｌｏｂａｌｓｏｌｕｔｉｏｎｏｆｔｈｅｉｎｉｔｉａｌ－ｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｆｏｒｔｈｅｆｏｌｌｏｗｉｎｇｎｏｎｌｉｎｅａｒｈｙｐｅｒｂｏｌｉｃｅｑｕａｔｉｏｎｕｔｔ＋Ａｕ＋｜ｕｔ｜αｕｔ＝ｆ（ｘ,ｔ）　　　　　ｉｎΩ×Ｒ＋,（１）ｕ（ｘ,０）＝ｕ０（ｘ）,ｕｔ（ｘ,０）＝ｕ１（ｘ）　　ｘ∈Ω,（２）ｕ（ｘ,ｔ）＝０　　　　　　　　　　　　（ｘ,ｔ…     

函数f（x）＝｜sinβx／sinαx｜（α，β＞0）的周期性

函数ｆ（ｘ）＝｜ｓｉｎβｘ／ｓｉｎαｘ｜（α，β＞０）的周期性曾丕刚陕西省镇安县中学７１１５００定理函数ｆ（ｘ）：为周期函数的充要条件是为周期函数，Ｌ为周期．（２）若ｆ（Ｘ）是周期函数，设ｔ（ｔ＞０）是它的周期，则ｆ（ｘ＋ｔ）＝ｆ（Ｘ），即在定义域内...     

非强制性Kirchhoff方程

本文讨论下述非齐次Ｋｉｒｃｈｈｏｆ方程的Ｃａｕｃｈｙ问题ｕｔｔ－ｍ∫Ｒｎ｜Δｕ｜２ｄｘΔｕ＝ｆ（ｔ，ｘ），ｕ（０，ｘ）＝ｕ０（ｘ），ｕｔ（０，ｘ）＝ｕ１（ｘ），其中ｍ（ｒ）∈Ｃ１［０，＋∞）且ｍ（ｒ）＞０．在初值和右端项“小”的条件下，我们获得了此问题整体解的存在唯一性     

二维带形无界区域中Navier—Stokes方程整体吸引子及其维数估计

该文讨论二维无界带形区域中Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程（Ⅰ）｛ｕｔ－△ｕ＋ｕｉэｕэｘｉ＝－△ｐ＋ｆ（ｘ，ｔ）∈Ω×Ｒ＋（１）ｄｉｖｕ＝０（２）ｕ（Ｘ，ｔ）∈（Ｈ＾１０（Ω）ｆｏｒ ｔ〉０（３）ｕ（ｘ，０）＝ｕ０（ｘ）∈Ｈ（４）其中Ω＝（０，ｄ）×Ｒ，ｄ〉０为一常数，ｕ与ｐ为未知量，其中ｕ＝（ｕ１，ｕ２）为速度场，ｐ表示压力。我们证明了当ｕ０∈Ｈ，ｆ∈Ｖ且ｆ「ｌｏｇ（ｅ＋│ｘ│＾２）」＾１２∈Ｌ     

巧用函数单调性解赛题

函数ｆ（ｘ）在区间［ａ,ｂ］上单调增加（或单调减少）,又ｃ、ｄ∈［ａ,ｂ］上,若ｆ（ｃ）＝ｆ（ａ）,则有ｃ＝ｄ．１　求代数式的值例１　已知ｘ、ｙ∈［－π４,π４］,ａ∈Ｒ,且　ｘ３＋ｓｉｎｘ－２ａ＝０４ｙ３＋ｓｉｎｙｃｏｓｙ＋ａ＝０则ｃｏｓ（ｘ＋２ｙ）＝　　．（１９９４年全国高中数学竞赛题）解　由已知条件,可得　　ｘ３＋ｓｉｎｘ＝２ａ（－２ｙ）３＋ｓｉｎ（－２ｙ）＝２ａ故可设函数ｆ（ｔ）＝ｔ３＋ｓｉｎｔ,则有ｆ（ｘ）＝ｆ（－２ｙ）＝２ａ．由于函数ｆ（ｔ）＝ｔ３＋ｓｉｎｔ,在［－π２,π２］上是单…     

二阶奇异边值问题两个非负解的存在性

利用拓扑度理论,给出了边值问题ｕ″（ｔ）＋λａ（ｔ）ｆ（ｕ（ｔ））＝０,０〈ｔ〈１,ａｕ（０）－βｕ’（０）＝０,γｕ（１）＋δｕ’（１）＝０两个非负解的存在性结果,这里允许ａ在ｔ＝０和ｔ＝１处有奇性。     

一类反应扩散方程解的渐近性质及其应用

本文讨论问题 ｕｔ＝Ａｕｘｘ＋ｆ（ｘ，ｔ，ｕ）， ｕｘ｜ｘ＝０＝０，ｕｘ｜ｘ＝１＝０， ｕ｜ｔ＝０＝ｕ０（ｘ）。 的解的渐近性质，将参考文献［１］的Ｌ＾２范数估计     

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.     

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     

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.     

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).     

非线性椭圆方程自由边值问题球对称解的存在性

给出了如下的非线性椭圆方程自由边值问题-Δu=λu+(1+ε)u+p,x∈B Rn,u|Ω=μ,∫Ωnu=-M(1)在C[0,1]中的球对称解的存在性.并得到比上述问题更一般的非线性椭圆方程自由边值问题-Δu=h(u),x∈B Rn,u|Ω=μ,∫Ωun=-M,在C[0,1]中的球对称解的存在性,其中B为Rn中的单位球,p>1,λ>0,μ<0,M>0,ε>0;λ,μ,M,ε均为常数,n为正整数.     

On Nonlinear Eigen-problems of Quasi-linear Elliptic Operators

In this paper, we study the following Eigen-problem {-\frac{∂}{∂x_i}(a_{ij}(x, u)\frac{∂u}{∂x_j}) + \frac{1}{2}a_{iju}(x,u)\frac{∂u}{∂x_i}\frac{∂u}{∂x_j} + h(x)u = μμ\frac{n+2}{n-2} \quad in Ω \qquad (0.1) u = 0 \quad on ∂Ω u > 0 \quad in Ω ⊂ R^n under some assumptions. First. we minimize I(u) = \frac{1}{2}∫_Ωa_{ij}(x, u)\frac{∂u}{∂x_i}\frac{∂u}{∂x_j} + h(x)u² over E_α = {u ∈ H¹_0(Ω); ∫_Ωu^α = 1} ( 2 < α < N = \frac{2n}{n-2}) to give a H¹_0-solution U_α of the perturbation problems of (0.1). Since I is not differentiable in H¹_0(Ω), the key point is the estimate of U_α. Then, we derive local uniform bounds of (U_α) and give a 'bad' solution of (0.1). Last, we remove the singular points of the 'bad' solution to obtain a solution of (0.1), our result is a extension of that of Brezis & Nirenberg.     

TIME OPTIMAL BOUNDARY CONTROL FOR SYSTEMS GOVERNED BY PARABOLIC EQUATIONS

In this paper we consider the systems governed, by parabolioc equations \[\frac{{\partial y}}{{\partial t}} = \sum\limits_{i,j = 1}^n {\frac{\partial }{{\partial {x_i}}}} ({a_{ij}}(x,t)\frac{{\partial y}}{{\partial {x_j}}}) - ay + f(x,t)\] subject to the boundary control \[\frac{{\partial y}}{{\partial {\nu _A}}}{|_\sum } = u(x,t)\] with the initial condition \[y(x,0) = {y_0}(x)\] We suppose that U is a compact set but may not be convex in \[{H^{ - \frac{1}{2}}}(\Gamma )\], Given \[{y_1}( \cdot ) \in {L^2}(\Omega )\] and d>0, the time optimal control problem requiers to find the control \[u( \cdot ,t) \in U\] for steering the initial state {y_0}( \cdot )\] the final state \[\left\| {{y_1}( \cdot ) - y( \cdot ,t)} \right\| \le d\] in a minimum, time. The following maximum principle is proved: Theorem. If \[{u^*}(x,t)\] is the optimal control and \[{t^*}\] the optimal time, then there is a solution to the equation \[\left\{ {\begin{array}{*{20}{c}} { - \frac{{\partial p}}{{\partial t}} = \sum\limits_{i,j = 1}^n {\frac{\partial }{{\partial {x_i}}}({a_{ji}}(x,t)\frac{{\partial p}}{{\partial {x_j}}}) - \alpha p,} }\{\frac{{\partial p}}{{\partial {\nu _{{A^'}}}}}{|_\sum } = 0} \end{array}} \right.\] with the final condition \[p(x,{t^*}) = {y^*}(x,{t^*}) - {y_1}(x)\], such that \[\int_\Gamma {p(x,t){u^*}} (x,t)d\Gamma = \mathop {\max }\limits_{u( \cdot ) \in U} \int_\Gamma {p(x,t)u(x)d\Gamma } \]     

Multiple periodic solutions of a superlinear forced wave equation

We study the existence of forced vibrations of nonlinear wave equation: (*) $$\begin{array}{*{20}c} {u_{tt} - u_{xx} + g(u) = f(x,t),} & {(x,t) \in (0,\pi ) \times R,} \\ {\begin{array}{*{20}c} {u(0,t) = u(\pi ,t) = 0,} \\ {u(x,t + 2\pi ) = u(x,t),} \\ \end{array} } & {\begin{array}{*{20}c} {t \in R,} \\ {(x,t) \in (0,\pi ) \times R,} \\ \end{array} } \\ \end{array}$$ whereg(ξ)∈C(R,R)is a function with superlinear growth and f(x, t) is a function which is 2π-periodic in t. Under the suitable growth condition on g(ξ), we prove the existence of infinitely many solution of (*) for any given f(x, t).     

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.     

Optimal Transportation for Generalized Lagrangian

This paper deals with the optimal transportation for generalized Lagrangian L = L(x, u, t), and considers the following cost function: c(x, y) = inf x(0)=x x(1)=y u∈U∫_0~1 L(x(s), u(x(s), s), s)ds, where U is a control set, and x satisfies the ordinary equation x(s) = f(x(s), u(x(s), s)).It is proved that under the condition that the initial measure μ0 is absolutely continuous w.r.t. the Lebesgue measure, the Monge problem has a solution, and the optimal transport map just walks along the characteristic curves of the corresponding Hamilton-Jacobi equation:V_t(t, x) + sup u∈UV_x(t, x), f(x, u(x(t), t), t)-L(x(t), u(x(t), t), t) = 0,V(0, x) = Φ0(x).