EXISTENCE AND NONEXISTENCE OF GLOBAL SOLUTIONS FOR A SEMI-LINEAR HEAT EQUATION WITH FRACTIONAL LAPLACIAN

In this paper,we are concerned with the existence and non-existence of global solutions of a semi-linear heat equation with fractional Laplacian.We obtain some extension of results of Weissler who considered the case α = 1,and h ≡ 1.     

随机发展方程适度解的整体存在唯一性

本文考虑了如下Ｈｉｌｂｅｒｔ空间中半线性随机发展方程（１）的Ｃａｕｃｈｙ问题，在两组不同的条件下，分别得到了该问题的适度解的整体存在唯一性。     

EXISTENCE OF SOLUTION AND APPROXIMATE CONTROLLABILITY OF A SECOND-ORDER NEUTRAL STOCHASTIC DIFFERENTIAL EQUATION WITH STATE DEPENDENT DELAY

This paper has two sections which deals with a second order stochastic neutral partial differential equation with state dependent delay. In the first section the existence and uniqueness of mild solution is obtained by use of measure of non-compactness. In the second section the conditions for approximate controllability are investigated for the distributed second order neutral stochastic differential system with respect to the approximate controllability of the corresponding linear system in a Hilbert space. Our method is an extension of co-author N. Sukavanam's novel approach in [22]. Thereby, we remove the need to assume the invertibility of a controllability operator used by authors in [5], which fails to exist in infinite dimensional spaces if the associated semigroup is compact. Our approach also removes the need to check the invertibility of the controllability Gramian operator and associated limit condition used by the authors in [20], which are practically difficult to verify and apply. An example is provided to illustrate the presented theory.     

一类带乘性噪声随机分数阶微分方程Euler方法的弱收敛性与弱稳定性

本文主要在带加性噪声随机分数阶微分方程的基础上,研究了一类更为困难的带乘性噪声随机分数阶微分方程Euler方法的弱收敛性与弱稳定性,并得到了类似的结论.首先构造了数值求解带乘性噪声随机分数阶微分方程的Euler方法,然后证明当分数阶α满足0α1/2时,该方法是1/2-α阶弱收敛的和弱稳定的,文末数值试验的结果验证了理论结果的正确性.     

PRECISE MOMENT ASYMPTOTICS FOR THE STOCHASTIC HEAT EQUATION OF A TIME-DERIVATIVE GAUSSIAN NOISE

This article establishes the precise asymptotics Eu~m(t, x)(t →∞ or m →∞) for the stochastic heat equation ?u/?t(t, x) =1/2?u(t, x) + u(t, x)(t, x)?W/?t(t, x) with the time-derivative Gaussian noise W?/?t(t, x) that is fractional in time and homogeneous in space.     

一类生物模型方程的数值解及其分析

本文利用小波 -Galerkin方法 ,求解生物领域的一模型方程 :一类带有小位移的线性二阶微分 -差分方程 (DDE) ,并对解的边界层性质进行数值探讨。结果表明 ,当小位移增加但仍保持很小时 ,解的边界层结构发生改变 ,甚至遭破坏     

EXISTENCE AND UNIQUENESS OF SOLUTIONS TO STOCHASTIC DIFFERENTIAL EQUATION WITH RANDOM COEFFICIENTS

This paper mainly deals with a stochastic differential equation (SDE) with random coefficients. Sufficient conditions which guarantee the existence and uniqueness of solutions to the equation are given.     

NONLINEAR STOCHASTIC HEAT EQUATION DRIVEN BY SPATIALLY COLORED NOISE:MOMENTS AND INTERMITTENCY

In this article, we study the nonlinear stochastic heat equation in the spatial domain R~d subject to a Gaussian noise which is white in time and colored in space. The spatial correlation can be any symmetric, nonnegative and nonnegative-definite function that satisfies Dalang's condition. We establish the existence and uniqueness of a random field solution starting from measure-valued initial data. We find the upper and lower bounds for the second moment. With these moment bounds, we first establish some necessary and sufficient conditions for the phase transition of the moment Lyapunov exponents, which extends the classical results from the stochastic heat equation on Z~d to that on R~d. Then,we prove a localization result for the intermittency fronts, which extends results by Conus and Khoshnevisan [9] from one space dimension to higher space dimension. The linear case has been recently proved by Huang et al [17] using different techniques.     

二维分数阶变系数对流-扩散方程的数值解

1 引 言 在古典的微分方程中,利用阶数为实数的分数阶导数代替整数阶导数,能更好地模拟许多物理现象.特别是具有复杂结构的多孔介质中的反常扩散问题.     

NUMERICAL SOLUTION OF THE SPACE FRACTIONAL DIFFERENTIAL EQUATION

In this paper, a space fractional differential equation is considered. The equation is obtained from the parabolic equation containing advection, diffusion and reaction terms by replacing the second order derivative in space by a fractional derivative in space of order. An implicit finite difference approximation for this equation is presented. The stability and convergence of the finite difference approximation are proved. A fractional-order method of lines is also presented. Finally, some numerical results are given.     

EXISTENCE TO FRACTIONAL CRITICAL EQUATION WITH HARDY-LITTLEWOOD-SOBOLEV NONLINEARITIES

In this paper,we consider the following new Kirchhoff-type equations involving the fractional p-Laplacian and Hardy-Littlewood-Sobolev critical nonlinearity:(a ...     

一类带有反应项的趋化性方程组解的存在唯一性

本文研究了主方程中带有反应项的Keller-Segel趋化性模型.利用截断函数、半群算子和不动点定理,获得了范数估计以及模型局部解的存在唯一性.     

具有时滞的n维Lienard型方程的调和解

本文讨论具有周期扰动的ｎ维时滞Ｌｉｅｎａｒｄ型方程ｐ（ｔ）的调和解．给出了存在调和解的若干充分条件,推广了ＧｅＷｅｉｇａｏ（１９９１）文中的相关结果．     

EXISTENCE AND UNIQUENESS OF THE SOLUTION FOR A CLASS OF NONLINEAR FRACTIONAL DIFFERENTIAL SYSTEM

In this paper we prove the existence and uniqueness of the solution for a class of nonlinear fractional differential system, and investigate the dependence of the solution on the orderαi.     

EXISTENCE UNIQUENESS AND DECAY OF SOLUTION FOR FRACTIONAL BOUSSINESQ APPROXIMATION

The Boussinesq approximation finds more and more frequent use in geologi- cal practice. In this paper, the asymptotic behavior of solution for fractional Boussinesq approximation is studied. After obtaining some a priori estimates with the aid of eommu- tator estimate, we apply the Galerkin method to prove the existence of weak solution in the case of periodic domain. Meanwhile, the uniqueness is also obtained. Because the results obtained are independent of domain, the existence and uniqueness of the weak solution for Cauchy problem is also true. Finally, we use the Fourier splitting method to prove the decay of weak solution in three cases respectively.     

无穷时滞Logistic方程的概周期解存在定理

本文讨论一类具无穷时滞Logistic系统,得到了关于该系统的一致持久性结论,由此改进文[3]中概周期解存在定理,同时解决了该文中的一个猜想.     

EXISTENCE AND UNIQUENESS OF NON-TRIVIAL SOLUTION OF PARABOLIC p-LAPLACIAN-LIKE DIFFERENTIAL EQUATION WITH MIXED BOUNDARIES

One parabolic p-Laplacian-like differential equation with mixed boundaries is studied in this paper,where the item (u)/(t) in the corresponding studies is replaced by α((u)/(t)),which makes it more general.The sufficient condition of the existence and uniqueness of non-trivial solution in L~2(0,T;L~2(Ω)) is presented by employing the techniques of splitting the boundary problems into operator equation.Compared to the corresponding work,the restrictions imposed on the equation are weaken and the proof technique is simplified.It can be regarded as the extension and complement of the previous work.     

EXISTENCE AND NONEXISTENCE OF GLOBAL SOLUTION OF NONLINEAR PARABOLIC EQUATION WITH NONLINEAR BOUNDARY CONDITION

ＥＸＩＳＴＥＮＣＥＡＮＤＮＯＮＥＸＩＳＴＥＮＣＥＯＦＧＬＯＢＡＬＳＯＬＵＴＩＯＮＯＦＮＯＮＬＩＮＥＡＲＰＡＲＡＢＯＬＩＣＥＱＵＡＴＩＯＮＷＩＴＨＮＯＮＬＩＮＥＡＲＢＯＵＮＤＡＲＹＣＯＮＤＩＴＩＯＮ￥ＷＵＹＯＮＧＨＵＩ；ＷＡＮＧＭＩＮＧＸＩＮＡｂｓｔｒ...     

EXISTENCE OF INFINITE ENERGY SOLUTION TO THE INELASTIC BOLTZMANN EQUATION WITH EXTERNAL FORCE

In this paper,the Cauchy problem for the inelastic Boltzmann equation with external force is considered in the case of initial data with infinite energy.More precisely,under the assumptions on the bicharacteristic generated by external force,we prove the global existence of solution for small initial data compared to the local Maxwellian exp{ p|x v|2},which has infinite mass and energy.     

无穷时滞Logistic方程的概周期解存在定理

本文讨论一类具无穷时滞Logistic系统,得到了关于该系统的一致持久性结论,由此改进文[3]中概周期解存在定理,同时解决了该文中的一个猜想.