SUFFICIENT AND NECESSARY CONDITIONS ON THE EXISTENCE AND ESTIMATES OF BOUNDARY BLOW-UP SOLUTIONS FOR SINGULAR p-LAPLACIAN EQUATIONS

Let?denote a smooth,bounded domain in R^(N)(N≥2).Suppose that g is a nondecreasing C^(1)positive function and assume that b(x)is continuous and nonnegative inΩ,and that it may be singular on■Ω.In this paper,we provide sufficient and necessary conditions on the existence of boundary blow-up solutions to the p-Laplacian problem△_(p)u=b(x)g(u)for x∈Ω,u(x)→+∞as dist(x,■Ω)→0.The estimates of such solutions are also investigated.Moreover,when b has strong singularity,the nonexistence of boundary blow-up(radial)solutions and infinitely many radial solutions are also considered.

ON THE NECESSARY AND SUFFICIENT CONDITIONS TO SOLVE A HEAT EQUATION WITH GENERAL ADDITIVE GAUSSIAN NOISE

In this note, we consider stochastic heat equation with general additive Gaussian noise. Our aim is to derive some necessary and sufficient conditions on the Gaussian noise in order to solve the corresponding heat equation. We investigate this problem invoking two differen t met hods, respectively, based on variance compu tations and on pat h-wise considerations in Besov spaces. We are going to see that, as anticipated, both approaches lead to the same necessary and sufficient condition on the noise. In addition, the path-wise approach brings out regularity results for the solution.

Network Decomposition and Maximum Independent Set Part Ⅰ:Theoretic Basis

The structure and characteristics of a connected network are analyzed, and a special kind of sub-network, which can optimize the iteration processes, is discovered. Then, the sufficient and necessary conditions for obtaining the maximum independent set are deduced. It is found that the neighborhood of this sub-network possesses the similar characters, but both can never be allowed incorporated together. Particularly, it is identified that the network can be divided into two parts by a certain style, and then both of them can be transformed into a pair sets network, where the special sub-networks and their neighborhoods appear alternately distributed throughout the entire pair sets network. By use of this characteristic, the network decomposed enough without losing any solutions is obtained. All of these above will be able to make well ready for developing a much better algorithm with polynomial time bound for an odd network in the the application research part of this subject.

ON THE BELTRAMI FLOWS

To illustrate the general properties of the Beltrami flows,a new sufficient and necessary condi- tion of the general solution of the Beltrami flows has been given,on whose basis some sufficient and necessary conditions and sufficient conditions in literature are unified.The advantages of the new expressions are that one may get solutions of Beltrami flows with constant proportional factors which satisfy some given asymptotic decaying conditions at infinity.Finally,we find a class of spherical vortices of Beltrami flows,where the mo- tion trajectories of the fluid particles are confined within a sphere or two adjacent concentric spherical shells.We find that the superpositions of the spherical vortices of the same order with different polar axes may produce chaotic trajectories.

On the Construction and Analysis of Finite Volume Element Schemes with Optimal L^(2) Convergence Rate

We provide a general construction method for a finite volume element(FVE)scheme with the optimal L^(2)convergence rate.The k-(k-1)-order orthogonal condition(generalized)is proved to be a sufficient and necessary condition for a k-order FVE scheme to have the optimal L^(2) convergence rate in 1D,in which the independent dual parameters constitute a(k-1)-dimension surface in the reasonable domain in k-dimension.In the analysis,the dual strategies in different primary elements are not necessarily to be the same,and they are allowed to be asymmetric in each primary element,which open up more possibilities of the FVE schemes to be applied to some complex problems,such as the convection-dominated problems.It worth mentioning that,the construction can be extended to the quadrilateral meshes in 2D.The stability and H^(1) estimate are proved for completeness.All the above results are demon-strated by numerical experiments.

Optimal vaccination strategy for a mean-field stochastic susceptible-infected-vaccinated system

The parameters of biological system may change under the influence of different states or state processes.The change of parameters can also affect the dynamic behaviors of epidemic disease.This paper presents a mean field stochastic susceptible-infectedvaccinated(SIV)epidemic model which parameters depend on the state process.The sufficient and necessary conditions of optimal control of the novel epidemic model are obtained by maximum principle.Finally,we perform representative numerical simulations to illustrate the theoretical results and further discuss the feasibility based on the hand,foot and mouth disease(HFMD)data in China.

MODIFIED LEAST SQUARE METHOD ON COMPUTING DIRICHLET PROBLEMS

The singularity theory of dynamical systems is linked to the numerical computation of boundary value problems of differential equations. It turns out to be a modified least square method for a calculation of variational problem defined on Ck(Ω), in which the base functions are polynomials and the computation of problems is transferred to compute the coefficients of the base functions. The theoretical treatment and some simple examples are provided for understanding the modification procedure of the methods. A modified least square method on difference scheme is introduced with a general matrix form of dynamical systems. We emphasize the simplicity of the algorithm and only use Euler algorithm to compute initial value problems of ODEs. A better algorithm is needed to reduce the stiffness of ODEs.