UNIQUENESS OF VISCOSITY SOLUTIONS OF FULLY NONLINEAR SECOND ORDER PARABOLIC PDE''''S

This paper investigates the maximum principle for viscosity solutions of fully nonlinear, second order parabolic, possibly degenerate partial differential equations. Using this maximum principle the anthor prove that viscosity solutions of initial and bo unoary value problem for parabolic equations are unique.     

Discrete maximum principle for the finite element solution of linear non-stationary diffusion–reaction problems

The maximum principle is one of the basic characteristic properties of solutions of second order partial differential equations of parabolic (and elliptic) types. The preservation of this property for solutions of corresponding discretized problems is a very natural requirement in reliable and meaningful numerical modelling of various real-life phenomena (heat conduction, air pollution, etc.). In the present paper we analyse a full discretization of a quite general class of linear parabolic equations and present sufficient conditions for the validity of a discrete analogue of the maximum principle in the case when bilinear finite elements are used for discretization in space.     

一类高阶奇异边值问题正解存在的充分必要条件

利用上下解方法和极大值原理给出了一类高阶微分方程的奇异边值问题正解存在的充分必要条件．     

一致抛物型方程广义解的弱最大值原理和唯一性定理

Trudinger和Gilbarg—Trudinger对椭园型方程的广义解推广了古典的最大值原理,唯一性定理也有新发展。现在我们把结果推广到一致抛物型方程的第一边值问题。 设Ω是n维欧氏空间E~n中的有界域,Ω为其边界,Q=Ω×(O,T),T是有限值。用(Q)记空间W_2~1(Q)的子空间,其函数在意义下满足如下边界条件: u(x,0)=0,x∈Ω和u(x,t)=0,x∈Ω,t∈(0,T)。 在Q考虑下面形状的方程     

The Phase Field Equations and Gradient Flows of Marginal Functions

We prove existence of solutions to the initial-boundary value problem for the quasistationary phase field equations with a multidimensional order parameter. We show that these solutions satisfy the entropy maximum principle and the entropy production minimum principle. We obtain a new selection theorem for differential inclusion with multifunctions generated by weak differentials of marginal functions.     

A maximum principle for fourth order ordinary differential equations

We present a maximum principle for fourth order ordinary differential equations, based on a new approach involving counting of inflection points. We use our results to compute solutions of nonlinear equations describing static displacements of a uniform beam     

A Maximum Principle for Elliptic and Parabolic Equations with Oblique Derivative Boundary Problems

This paper prove a maximum principle for viscooity solutions of fully nonlinear, second order, uniformly elliptic and parabolic equations with oblique boundary value conditions.     

A priori bounds for degenerate and singular evolutionary partial integro-differential equations

We study quasilinear evolutionary partial integro-differential equations of second order which include time fractional p-Laplace equations of time order less than one. By means of suitable energy estimates and De Giorgi’s iteration technique we establish results asserting the global boundedness of appropriately defined weak solutions of these problems. We also show that a maximum principle is valid for such equations.     

Boundedness of weak solutions to evolutionary partial integro-differential equations with discontinuous coefficients

We investigate linear and quasilinear evolutionary partial integro-differential equations of second order which include time fractional evolution equations of time order less than one. By means of suitable energy estimates and De Giorgi's iteration technique we establish results asserting the global boundedness of appropriately defined weak solutions of these problems. We also show that a maximum principle holds for such equations.     

Positive Solutions of Fully Nonlinear Elliptic Equations on General Bounded Domains

We prove the refined ABP maximum principle, comparison principle, and related existence and uniqueness theorem for the positive solutions of the Dirichlet problems of second order fully nonlinear elliptic equations on arbitrary bounded domains.     

一类四阶奇异边值问题的正解存在的充分必要条件

利用上下解方法和极大值原理给出了一般边界条件下四阶微分方程的奇异迫值问题有C^2[0，1]和C^3[0，1]正解存在的充分必要条件．推广了韦忠礼(1999)的结果。     

拟线性椭圆型方程广义解的最大原理

在允许自由项关于解梯度的增长阶满足自然增长条件时，证明了拟线性椭圆型方程不恒等于常数的有界广义解成立解的最大值原理。     

On a Maximum Principle and Uniqueness Theorems for a System of First Order Equations of Mixed Type

A maximum principle for a system of first order equations of mixed type is established. The uniqueness theorems of solutions t.o the generalized Tricomi type problem and to the Frankl's problem are proved by the method of auxiliary functions.     

Backward stochastic Volterra integral equations and some related problems

Backward stochastic Volterra integral equations (BSVIEs, for short) are introduced. The existence and uniqueness of adapted solutions are established. A duality principle between linear BSVIEs and (forward) stochastic Volterra integral equations is obtained. As applications of the duality principle, a comparison theorem is proved for the adapted solutions of BSVIEs, and a Pontryagin type maximum principle is established for an optimal control of stochastic integral equations.     

PERIODIC BOUNDARY VALUE PROBLEM FOR SECOND ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS

This paper investigates the periodic boundary value problems for a class of second order functional differential equations. The monotone iterative technique and the maximum principle are applied to obtain the existence of maximal and minimal solutions.     

A method for constructing asymptotic expansions of bisingularly perturbed problems

In this paper we propose an analog of the method of boundary functions for constructing uniform asymptotic expansions of solutions to bisingularly perturbed problems. With the help of this method we construct uniform asymptotic expansions of solutions to the Dirichlet problem for bisingularly perturbed ordinary differential equations and elliptic equations of the second order. By the use of the maximum principle we obtain estimates for the remainder terms.     

Multi-point boundary value problem of first order impulsive functional differential equations

In this paper, we introduce a new definition of lower and upper solutions for boundary value problem of first order impulsive functional differential equations with nonlinear multi-point boundary conditions. By developing a new maximum principle and using the monotone iterative technique, we obtain the extremal solutions of the boundary value problem.     

电报方程双周期解的极大值原理与强正性估计及应用

本文讨论非线性电报方程u_(tt)-u_(xx)+cu_t=F(t,x,u),(t,x)∈R~2时空双2π周期解的存在性。改进了Ortega与Robles-Perez关于线性电报方程双周期解的极大值原理,应用新获得的极大值原理,推广了相应的上下解定理,并且加强了极大值原理的结论,建立了线性方程解的强正性估计,利用这个强正性估计及锥上的不动点定理获得了超线性电报方程及奇异电报方程正双周期解的存在性。     

Dissipative mechanism of a semilinear higher order parabolic equation in RN

It is known that the concept of dissipativeness is fundamental for understanding the asymptotic behavior of solutions to evolutionary problems. In this paper we investigate the dissipative mechanism for some semilinear fourth-order parabolic equations in the spaces of Bessel potentials and discuss some weak conditions that lead to the existence of a compact global attractor. While for second-order reaction–diffusion equations the dissipativeness mechanism has already been satisfactorily understood (see Arrieta et al. (2004), doi:10.1142/S0218202504003234 [7]), for higher order problems in unbounded domains it has not yet been fully developed. As shown throughout the paper, one of the main differences from the case of reaction–diffusion equations stems from the lack of a maximum principle. Thus we have to rely here on suitable energy estimates for the solutions. As in the case of second-order reaction–diffusion equations, we show here that both linear and nonlinear terms have to collaborate in order to produce dissipativeness. Thus, the dissipative mechanisms in second-order and fourth-order equations are similar, although the lack of a maximum principle makes the proofs more difficult and the results not as complete.Finally, we make essential use of the sharp results of Cholewa and Rodriguez-Bernal (2012), doi:10.1016/j.na.2011.08.022 [12], on linear fourth-order equations with a very large class of linear potentials.     

Positivity preserving property for a class of biharmonic elliptic problems

The lack of a general maximum principle for biharmonic equations suggests to study under which boundary conditions the positivity preserving property holds. We show that this property holds in general domains for suitable linear combinations of Dirichlet and Navier boundary conditions. The spectrum of this operator exhibits some unexpected features: radial data may generate nonradial solutions. These boundary conditions are also of some interest in semilinear equations, since they enable us to give explicit radial singular solutions to fourth order Gelfand-type problems.