Maximum modulus principle estimates for one dimensional fractional diffusion equation

We present scheme I for solving one-dimensional fractional diffusion equation with variable coefficients based on the maximum modulus principle and two Grunwald approxima- tions. Scheme II is obtained by using classic Crank-Nicolson approximations in order to improve the time convergence. Schemes are proved to be unconditionally stable and second-order accuracy in spatial grid size for the problem with order of fractional derivative belonging to [（√17- 1）/2, 2] using the maximum modulus principle. A numerical example is given to confirm the theoretical analysis result.

A RISK-SENSITIVE STOCHASTIC MAXIMUM PRINCIPLE FOR OPTIMAL CONTROL OF JUMP DIFFUSIONS AND ITS APPLICATIONS

A stochastic maximum principle for the risk-sensitive optimal control prob- lem of jump diffusion processes with an exponential-of-integral cost functional is derived assuming that the value function is smooth, where the diffusion and jump term may both depend on the control. The form of the maximum principle is similar to its risk-neutral counterpart. But the adjoint equations and the maximum condition heavily depend on the risk-sensitive parameter. As applications, a linear-quadratic risk-sensitive control problem is solved by using the maximum principle derived and explicit optimal control is obtained.

GLOBAL STABILITY OF TRAVELING WAVEFRONTS FOR NONLOCAL REACTION-DIFFUSION EQUATIONS WITH TIME DELAY

This paper is concerned with the stability of traveling wavefronts for a population dynamics model with time delay. Combining the weighted energy method and the comparison principle, the global exponential stability of noncritical traveling wavefronts （waves with speeds c 〉 c＊, where c=c＊ is the minimal speed） is established, when the initial perturbations around the wavefront decays to zero exponentially in space as x → -∞, but it can be allowed arbitrary large in other locations, which improves the results in[9, 18, 21].

A Mean-Field Stochastic Maximum Principle for Optimal Control of Forward-Backward Stochastic Differential Equations with Jumps via Malliavin Calculus

This paper considers a mean-field type stochastic control problem where the dynamics is governed by a forward and backward stochastic differential equation (SDE) driven by Lévy processes and the information available to the controller is possibly less than the overall information. All the system coefficients and the objective performance functional are allowed to be random, possibly non-Markovian. Malliavin calculus is employed to derive a maximum principle for the optimal control of such a system where the adjoint process is explicitly expressed.

Global existence and blow up of solutions for two classes of reaction diffusion systems with two nonlinear source terms in bounded domain

In this paper we deal with the initial boundary value problem for two classes of reaction diffusion systems with two source terms in bounded domain. Under some assumptions on the exponents and the initial data, applying the comparison principle, the maximum prin- ciple and the supersolution-subsolution method, we prove the global existence and blow up of solutions. We also establish some upper blow up rates.

Nonlinear Singularly Perturbed Problems for Reaction Diffusion Equations with Two Parameters and Boundary Perturbation

A class of nonlinear singularly perturbed initial boundary value problems for reaction diffusion equations with two parameters and boundary perturbation were considered.Under suitable conditions,the existence,uniqueness and asymptotic behavior of solutions for the initial boundary value problems were studied.An example was also given to illustrate our main results.

Necessary Maximum Principle of Stochastic Optimal Control with Delay and Jump Diffusion

In this paper, we have studied the necessary maximum principle of stochastic optimal control problem with delay and jump diffusion.

Wave equations and reaction-diffusion equations with several nonlinear source terms

The initial boundary value problem of wave equations and reaction-diffusion equations with several nonlinear source terms in a bounded domain is studied by potential well method. The invarianee of some sets under the flow of these problems and the vacuum isolation of solutions are obtained by introducing a family of potential wells. Then the threshold result of global existence and nonexistence of solutions are given. Finally, the problem with critical initial conditions are discussed.

UPWIND SPLITTING SCHEME FOR CONVECTION-DIFFUSION EQUATIONS

WT5,5”BX] A new class of numerical schemes is proposed to solve convection diffusion equations by combining the upwind technique and the method of operator splitting. For every time step, the multi dimensional approximation is performed in several independent directions alternatively, while the upwind technique is applied to treat the convection term in every individual direction. This scheme possesses maximum principle. Stability and convergence are analysed by energy method.[WT5,5”HZ]

The Corrected Finite Volume Element Methods for Diffusion Equations Satisfying Discrete Extremum Principle

In this paper,we correct the finite volume element methods for diffusion equations on general triangular and quadrilateral meshes.First,we decompose the numerical fluxes of original schemes into two parts,i.e.,the principal part with a twopoint flux structure and the defective part.And then with the help of local extremums,we transform the original numerical fluxes into nonlinear numerical fluxes,which can be expressed as a nonlinear combination of two-point fluxes.It is proved that the corrected schemes satisfy the discrete strong extremum principle without restrictions on the diffusion coefficient and meshes.Numerical results indicate that the corrected schemes not only satisfy the discrete strong extremum principle but also preserve the convergence order of the original finite volume element methods.

Stochastic Maximum Principle for Forward-Backward Regime Switching Jump Diffusion Systems and Applications to Finance

The authors prove a sufficient stochastic maximum principle for the optimal control of a forward-backward Markov regime switching jump diffusion system and show its connection to dynamic programming principle. The result is applied to a cash flow valuation problem with terminal wealth constraint in a financial market. An explicit optimal strategy is obtained in this example.

Exponential Time Differencing Schemes for the Peng-Robinson Equation of State with Preservation of Maximum Bound Principle

The Peng-Robison equation of state,one of the most extensively applied equations of state in the petroleum industry and chemical engineering,has an excel-lent appearance in predicting the thermodynamic properties of a wide variety of ma-terials.It has been a great challenge on how to design numerical schemes with preser-vation of mass conservation and energy dissipation law.Based on the exponential time difference combined with the stabilizing technique and added Lagrange multi-plier enforcing the mass conservation,we develop the efficientfirst-and second-order numerical schemes with preservation of maximum bound principle(MBP)to solve the single-component two-phase diffuse interface model with Peng-Robison equation of state.Convergence analyses as well as energy stability are also proven.Several two-dimensional and three-dimensional experiments are performed to verify these theo-retical results.

EVANS FUNCTIONS AND INSTABILITY OF A STANDING PULSE SOLUTION OF A NONLINEAR SYSTEM OF REACTION DIFFUSION EQUATIONS

In this paper, we consider a nonlinear system of reaction diffusion equa- tions arising from mathematical neuroscience and two nonlinear scalar reaction diffusion equations under some assumptions on their coefficients. The main purpose is to couple together linearized stability criterion （the equivalence of the nonlinear stability, the linear stability and the spectral sta- bility of the standing pulse solutions） and Evans functions to accomplish the existence and instability of standing pulse solutions of the nonlinear system of reaction diffusion equations and the nonlinear scalar reaction diffusion equa- tions. The Evans functions for the standing pulse solutions are constructed explicitly.

QUENCHING PROBLEMS OF DEGENERATE FUNCTIONAL REACTION-DIFFUSION EQUATION

This paper is concerned with the quenching problem of a degenerate functional reaction-diffusion equation. The quenching problem and global existence of solution for the reaction-diffusion equation are derived and, some results of the positive steady state solutions for functional elliptic boundary value are also presented.

THE SINGLE-POINT QUENCHING OF NONLINEAR DEGENERATE FUNCTIONAL REACTION-DIFFUSION EQUATION

This article is concerned with the quenching phenomena of the nonlinear degenerate functional reaction-diffusion equation. Some results are obtained on the single-point quenching and the uniqueness of quenching.

一类反应扩散SIQR传染病模型的最优控制

为探究疾病发生时如何更快地控制疾病、避免疾病的扩散和流行,研究了一类反应扩散的易感-感染-隔离-恢复(susceptible-infectious-quarantined-recovered,SIQR)传染病模型的最优控制问题。首先利用截断方法证明状态方程解y=(S,I,Q,R)^(T)的存在性,其次利用极小化序列证明最优控制对(S^(*),I^(*),Q^(*),R^(*),u^(*))的存在性,最后利用拉格朗日方法得到一阶必要条件以及最优控制u^(*)的表达形式。结果表明:当控制u=u^(*)为最优控制时,给定的目标泛函达到最小值。该研究在偏微分方程传染病模型的最优控制方面具有重要意义。

Positivity-Preserving Numerical Methods for Belousov-Zhabotinsky Reaction

The existence of positive solutions to the system of ordinary differential equations related to the Belousov-Zhabotinsky reaction is established. The key idea is to use a new successive approximation of solutions, ensuring its positivity. To obtain the positivity and invariant region for numerical solutions, the system is discretized as difference equations of explicit form, employing operator splitting methods with linear stability conditions. Algorithm to solve the alternate solution is given.

一类黑色素瘤耐药机制交叉扩散方程组模型解的存在性

研究了检查点抑制剂靶向治疗人类黑色素瘤的数学模型。该模型由12个耦合的反应扩散方程组成,模型具有不连续项且包含自由边界条件。将自由边界问题转化为固定边界问题,并利用抛物方程的Lp理论和Schauder不动点定理,结合函数逼近的方法,得到了该数学模型全局弱解的存在性。

BEHAVIOR OF SOLUTIONS FOR A MODEL IN HETEROGENEOUS CATALYTIC REACTION-DIFFUSION

The purpose of this paper is t0 investigate an irreversible model in the kine-tics of heterogeneous cataIytic reaction-diffusion. The existence, uniqueness andlarge-time behavior of solutions are proved. Particularly, the result shows thatthe reaction ceare in nnite time provided there is some kinds of absorption.

带记忆项半线性反应扩散方程的强吸引子

本文讨论带衰退记忆项的非线性反应扩散方程整体强解的长时间行为.首先,利用控制收敛原理及解的正则性证明系统解半群为积空间H^(1)_(0)(Ω)×L^(2)_(μ)(R;D(A))上的压缩半群,由此得到解半群的渐近紧性;然后,证明积空间上全局吸引子A的存在性和正则性.值得注意的是非线性项f满足任意阶多项式增长条件,并且A⊂D(A)×L^(2)_(μ)(R;D(A)).