三维种群生态动力系统的参数识别

研究具有反应一扩散现象的三维种群生态动力系统的参数识别问题，依该系统正问题解的性质，建立了参数识别的数学模型；论证了系统正问题解关于待识别参数的连续依赖性与参数识别问题最优解的存在性。     

具有非局部边界条件的强耦合反应扩散方程组

考虑具有非局部边界条件的半线性强耦合反应扩散方程组的初值问题，利用上，下解方法和Ｌｅｒａｙ－Ｓｃｈａｕｄｅｒ不动点定理等，证明问题在适当条件下的光滑解的存在唯一性。     

关于HBV及其药物相互作用的反应扩散方程的参数识别

研究具有反应-扩散现象的HBV及其药物相互作用系统的参数识别问题,依该系统正问题解的性质,建立了参数识别问题的数学模型,论证了系统正问题解关于待识别参数的连续依赖性与参数识别问题最优解的存在性.     

一类带小参数反应——扩散型方程组的性态估计

得到了激光等离子能量交换模型研究中的一类反应－－扩散方程组的本解的存在性。并通过引进光滑符号函数对解析解的性态进行了估计,为数值方法的误差分析提供了理论依据。     

具有足够大 Ginzburg-Landau参数的一维Ginzburg-Landau超导方程组对称解的唯一性

本文证明了：当Ｇｉｎｚｂｕｒｇ－Ｌａｎｄａｕ参数足够大时，一维Ｇｉｎｚｂｕｒｇ－Ｌａｎｄａｕ超导方程组的对称解 是唯一的．该问题的难点在于所考虑的解具有“奇点”：也即，当Ｇｉｎｚｂｕｒｇ－Ｌａｎｄａｕ参数趋于无穷大 时，解的导数在这些点处趋于无穷．证明的关键是要得到解在这些奇点近旁的精细估计．     

一类反应扩散方程组的渐近行为

本文讨论了一类反应扩散方程组的渐进行为,证明了对任意ｕ〉０解在Ｃ＾ｖ（Ω）中收敛于常数平衡解,并推广了以往的结论。     

关于Fujita型反应扩散方程组的Cauchy问题

本文研究Ｆｕｊｉｔａ型反应扩散方程组ｕｔ－Δｕ＝α１｜ｕ｜ｑ１－１ｕ＋β１｜ｖ｜ｐ１－１ｖ，（ｘ∈ＲＮ，ｔ＞０），ｖｔ－Δｖ＝α２｜ｕ｜ｑ２－１ｕ＋β２｜ｖ｜ｐ２－１ｖ，ｕ（ｘ，０）＝ｕ０（ｘ）０，ｖ（ｘ，０）＝ｖ０（ｘ）０，（ｘ∈ＲＮ）Ｌｐ解的整体存在性和有限时间Ｂｌｏｗ ｕｐ问题．这里ｑｉ＞１，ｐｉ＞１（ｉ＝１，２），α１０，α２＞０，β１＞０，β２０，１ｐ＋∞．     

动力学方程组中参数识别模型及其优化性质

本文建立了动力学方程组中参数优化识别的数学模型.这是个含有动态约束的泛函极小化问题.应用线性算子的连续性,证明了该问题最优解的存在性,即参数的可识别性.最后依据弱Gateaux微分,给出并证明达到最优解的一个必要条件.     

三维Wigner-Poisson方程组的Cauchy问题

考虑三维 Ｗｉｇｎｅｒ－Ｐｏｉｓｓｏｎ方程组的 Ｃａｕｃｈｙ问题，将 ＷＰ问题转化为等价的 Ｓｃｈｒｏｄｉｎｇｅｒ－Ｐｏｉｓｓｏｎ问题．采用有限区域序列上的解的逼近方法，通过对逼近解建立与区域无关的先验估计，证明了 Ｃａｕｃｈｙ问题解的存在性、唯一性和逼近解的收敛性     

一类非线性反应扩散方程解的Blow—up问题

本文得用极大值原理研究一类非线性反应扩散方程在各种边界条件下解的Ｂｌｏｗ－ｕｐ问题，给出了整体解不存在的一系列定理，并得到了Ｂｌｏｗ－ｕｐ时间Ｔ的上界。     

Parameter Estimation with the Augmented Lagrangian Method for a Parabolic Equation

In this paper, we investigate the numerical identification of the diffusion parameters in a linear parabolic problem. The identification is formulated as a constrained minimization problem. By using the augmented Lagrangian method, the inverse problem is reduced to a coupled nonlinear algebraic system, which can be solved efficiently with the preconditioned conjugate gradient method. Finally, we present some numerical experiments to show the efficiency of the proposed methods, even for identifying highly discontinuous parameters.This work was partially supported by the Research Council of Norway, Grant NFR-128224/431.     

具有时滞的Volterra反应扩散差分方程的稳定性

本文考虑一类时滞Volterra反应扩散差分方程的初边值问题的正稳态解的稳定性。利用上下解方法和单调迭代方法得到了每一个解趋于方程的正稳态解的充分条件。     

Stability and attractivity of periodic solutions of parabolic systems with time delays

This paper is concerned with the existence, stability, and global attractivity of time-periodic solutions for a class of coupled parabolic equations in a bounded domain. The problem under consideration includes coupled system of parabolic and ordinary differential equations, and time delays may appear in the nonlinear reaction functions. Our approach to the problem is by the method of upper and lower solutions and its associated monotone iterations. The existence of time-periodic solutions is for a class of locally Lipschitz continuous reaction functions without any quasimonotone requirement using Schauder fixed point theorem, while the stability and attractivity analysis is for quasimonotone nondecreasing and mixed quasimonotone reaction functions using the monotone iterative scheme. The results for the general system are applied to the standard parabolic equations without time delay and to the corresponding ordinary differential system. Applications are also given to three Lotka-Volterra reaction diffusion model problems, and in each problem a sufficient condition on the reaction rates is obtained to ensure the stability and global attractivity of positive periodic solutions.     

Dead-core solutions for slightly non-isothermal diffusion-reaction problems with power-law kinetics

The paper deals with dead-core solutions to a non-isothermal reaction- diffusion problem with power-law kinetics for a single reaction that takes place in a catalyst pellet along with mass and heat transfer from the bulk phase to the outer pellet surface. The model boundary value problem for two coupled non-linear diffusion-reaction equations is solved using the semi-analytical method. The exact solutions are established under the assumption of a small temperature gradient in the pellet. The nonlinear algebraic expressions are derived for the critical Thiele modulus, dead-zone length, reactant concentration, and temperature profiles in catalyst pellets of planar geometry. The effects of the reaction order, Arrhenius number, energy generation function, Thiele modulus, and Biot numbers are investigated on the concentration and temperature profiles, dead-zone length, and critical Thiele modulus.     

Stability estimates of solutions to extremal problems for a nonlinear convection-diffusion-reaction equation

We consider an identification problem for a stationary nonlinear convection–diffusion–reaction equation in which the reaction coefficient depends nonlinearly on the concentration of the substance. This problem is reduced to an inverse extremal problem by an optimization method. The solvability of the boundary value problem and the extremal problem is proved. In the case that the reaction coefficient is quadratic, when the equation acquires cubic nonlinearity, we deduce an optimality system. Analyzing it, we establish some estimates of the local stability of solutions to the extremal problem under small perturbations both of the quality functional and the given velocity vector which occurs multiplicatively in the convection–diffusion–reaction equation.     

Global attractor of a coupled finite difference reaction diffusion system with delays

In the study of asymptotic behavior of solutions for reaction diffusion systems, an important concern is to determine whether and when the system has a global attractor which attracts all positive time-dependent solutions. The aim of this paper is to investigate the global attraction problem for a finite difference system which is a discrete approximation of a coupled system of two reaction diffusion equations with time delays. Sufficient conditions are obtained to ensure the existence and global attraction of a positive solution of the corresponding steady-state system. Applications are given to three types of Lotka-Volterra reaction diffusion models, where time-delays may appear in the opposing species.     

Numerical solutions of coupled systems of nonlinear elliptic equations

This article deals with numerical solutions of a general class of coupled nonlinear elliptic equations. Using the method of upper and lower solutions, monotone sequences are constructed for difference schemes which approximate coupled systems of nonlinear elliptic equations. This monotone convergence leads to existence‐uniqueness theorems for solutions to problems with reaction functions of quasi‐monotone nondecreasing, quasi‐monotone nonincreasing and mixed quasi‐monotone types. A monotone domain decomposition algorithm which combines the monotone approach and an iterative domain decomposition method based on the Schwarz alternating, is proposed. An application to a reaction‐diffusion model in chemical engineering is given. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 621–640, 2012     

An Asymptotic Analysis of a 2-D Model of Dynamically Active Compartments Coupled by Bulk Diffusion

A class of coupled cell–bulk ODE–PDE models is formulated and analyzed in a two-dimensional domain, which is relevant to studying quorum-sensing behavior on thin substrates. In this model, spatially segregated dynamically active signaling cells of a common small radius \(\epsilon \ll 1\) are coupled through a passive bulk diffusion field. For this coupled system, the method of matched asymptotic expansions is used to construct steady-state solutions and to formulate a spectral problem that characterizes the linear stability properties of the steady-state solutions, with the aim of predicting whether temporal oscillations can be triggered by the cell–bulk coupling. Phase diagrams in parameter space where such collective oscillations can occur, as obtained from our linear stability analysis, are illustrated for two specific choices of the intracellular kinetics. In the limit of very large bulk diffusion, it is shown that solutions to the ODE–PDE cell–bulk system can be approximated by a finite-dimensional dynamical system. This limiting system is studied both analytically, using a linear stability analysis and, globally, using numerical bifurcation software. For one illustrative example of the theory, it is shown that when the number of cells exceeds some critical number, i.e., when a quorum is attained, the passive bulk diffusion field can trigger oscillations through a Hopf bifurcation that would otherwise not occur without the coupling. Moreover, for two specific models for the intracellular dynamics, we show that there are rather wide regions in parameter space where these triggered oscillations are synchronous in nature. Unless the bulk diffusivity is asymptotically large, it is shown that a diffusion-sensing behavior is possible whereby more clustered spatial configurations of cells inside the domain lead to larger regions in parameter space where synchronous collective oscillations between the small cells can occur. Finally, the linear stability analysis for these cell–bulk models is shown to be qualitatively rather similar to the linear stability analysis of localized spot patterns for activator–inhibitor reaction–diffusion systems in the limit of long-range inhibition and short-range activation.     

Asymptotic convergence of solutions for Laplace reaction–diffusion equations

We study the initial–boundary value problem for a Laplace reaction–diffusion equation. After constructing local solutions by using the theory of abstract degenerate evolution equations of parabolic type, we show asymptotic convergence of bounded global solutions if they exist under the assumption that the reaction function is analytic in neighborhoods of their $\omega$-limit sets. Reduction of degenerate evolution equation to multivalued evolution equation enables us to use the theory of the infinite-dimensional Łojasiewicz–Simon gradient inequality.     

A numerical method for a system of singularly perturbed reaction–diffusion equations

A Dirichlet problem for a system of two coupled singularly perturbed reaction–diffusion ordinary differential equations is examined. A numerical method whose solutions converge pointwise at all points of the domain independently of the singular perturbation parameters is constructed and analysed. Numerical results are presented, which illustrate the theoretical results.