Determination temperature of a backward heat equation with time-dependent coefficients

We introduce the truncation method for solving a backward heat conduction problem with time-dependent coefficients. For this method, we give the stability analysis with new error estimates. Meanwhile, we investigate the roles of regularization parameters in these two methods. These estimates prove that our method is effective.     

由软件和硬件构成的串联可修计算机系统的适定性及渐近稳定性

讨论由软件和硬件构成的串联可修计算机系统的时间依赖解,运用C_0-半群理论及算子理论,证明系统的适定性和时间依赖非负解的存在唯一性.通过研究系统相应算子的谱特征,得到系统时间依赖解的渐近稳定性.     

The fundamental solution on manifolds with time-dependent metrics

In this article we prove the existence of a fundamental solution for the linear parabolic operator L(u) = (Δ − ∂/∂t − h)u, on a compact n-dimensional manifold M with a time-parameterized family of smooth Riemannian metrics g(t). Δ is the time-dependent Laplacian based on g(t), and h(x, t) is smooth. Uniqueness, positivity, the adjoint property, and the semigroup property hold. We further derive a Harnack inequality for positive solutions of L(u) = 0 on (M, g(t) on a time interval depending on curvature bounds and the dimension of M, and in the special case of Ricci flow, use it to find lower bounds on the fundamental solution of the heat operator in terms of geometric data and an explicit Euclidean type heat kernel.     

Asymptotic behavior of an age-structured population model with diffusion

A reaction-diffusion system with stage-structure is studied. We provide well-posedness of the model and prove that time-dependent solutions evolve either towards a positive equilibrium or to the trivial one. Under suitable conditions, a branch of positive equilibrium is shown to exist.     

Gradient flow for the Boltzmann entropy and Cheeger’s energy on time-dependent metric measure spaces

We introduce notions of dynamic gradient flows on time-dependent metric spaces as well as on time-dependent Hilbert spaces. We prove existence of solutions for a class of time-dependent energy functionals in both settings. In particular, in the case when each underlying space satisfies a lower Ricci curvature bound in the sense of Lott, Sturm and Villani, we provide time-discrete approximations of the time-dependent heat flows introduced in Kopfer and Sturm (Heat flows on time-dependent metric measure spaces and super-Ricci flows, 2017. arXiv:1611.02570).     

Ultracontractivity for Brownian motion with Markov switching

Consider the transition density functions for Brownian motion with two-state Markov switching. The characteristic functions for transition density functions are presented. Then, we show that the semigroup-associated Brownian motion with Markov switching is ultracontractive. And an explicit time-dependent upper bound for heat kernels are presented. Moreover, we prove that the Dirichlet form associated Brownian motion with Markov switching satisfies the Nash inequality.     

第二种服务可选的M/G/1排队模型的适定性

运用Hille-Yosida定理,Phillips定理与Fattorini定理证明第二种服务可选的M/G/1排队模型存在唯一的概率瞬态解.     

服务员强制休假的M/G/1排队模型的适定性

运用Hille-Yosida定理,Phillips定理与Fattorini定理证明服务员强制休假的M/G/1排队模型存在唯一的概率瞬态解.     

批量到达的、具有两种服务阶段和服务中断的重试排队系统的适定性

运用Hille-Yosida定理,Phillips定理与Fattorini定理证明批量到达、具有两种服务阶段和服务中断的重试排队系统存在唯一的、非负的、满足概率性质的时间依赖解.     

Study of the Norton-Hoff operator coupled with the evolutive heat equation

We are interested in the study of quasistatic visco-plastic flows with thermal effects. The fluid motion is governed by the incompressible Norton-Hoff model coupled with the time-dependent heat equation where the dissipated mechanical power is the source term. The viscosity of the fluid is modeled by the non-linear Arrhenius law. The well-posedness of each decoupled system is given. The optimal regularities of the heat solution and of the scale factor are supplied. A non-linear operator describing the stand coupling is provided. The existence of a solution to the considered problem is established. We prove the compactness result of the set solutions.     

具有热储备的可修复平行系统在由常规错误引起失效下的渐进稳定性

讨论了具有热储备和两个独立相同部件的平行系统在由常规错误引起失效下的渐进稳定性.首先,利用Banach空间的Volttera算子方程得到了非负动态解的存在唯一性;然后,利用强连续线性算子半群理论证明了系统正的动态解的存在唯一性,而由于初始值不在定义域内,故得到的是mild解.但在t＞0时系统古典解存在唯一,所以此时mild解即为古典解.最后,利用线性算子半群稳定性的结果,证明了该动态解在范数意义下收敛到稳态解,进而得到了系统的渐进稳定性.     

M/Mk,B/1算子的豫解集

首先运用Phillips定理和Fattorini定理证明M／M^k，^B／1排队模型概率瞬态解的存在唯一性，然后通过研究对应于M／M^k，^B／1排队模型的主算子的共轭算子的豫解集得到该主算子的豫解集：在虚轴上除了零点外其它所有点都属于该主算子的豫解集．     

An Upper Bound for Hessian Matrices of Positive Solutions of Heat Equations

We prove global and local upper bounds for the Hessian of log positive solutions of the heat equation on a Riemannian manifold. The metric is either fixed or evolved under the Ricci flow. These upper bounds seem to be the first general ones that match the well-known lower bounds which have been around for some time. As an application, we discover a local, time reversed Harnack type inequality for bounded positive solutions of the heat equation.     

Lagrange stability for impulsive Duffing equations

This work discusses the boundedness of solutions for impulsive Duffing equation with time-dependent polynomial potentials. By KAM theorem, we prove that all solutions of the Duffing equation with low regularity in time undergoing suitable impulses are bounded for all time and that there are many (positive Lebesgue measure) quasi-periodic solutions clustering at infinity. This result extends some well-known results on Duffing equations to impulsive Duffing equations.     

半导体器件瞬态模拟的对称正定混合元方法

提出具有对称正定特性的混合元格式求解非稳态半导体器件瞬态模拟问题。提出一个最小二乘混合元方法、一个新的具有分裂和对称正定性质的混合元格式和一个解经典混合元方程的对称正定失窃工格式求解电场位势和电场强度方程;提出一个最小二乘混合元格式求解关于电子与空穴浓度的非稳态对流扩散方程,浓度函数和流函数被同时求解;采用标准的有限元方法求解热传导方程。建立了误差分析理论。     

Differential Harnack estimates for backward heat equations with potentials under the Ricci flow

In this paper, we derive a general evolution formula for possible Harnack quantities. As a consequence, we prove several differential Harnack inequalities for positive solutions of backward heat-type equations with potentials (including the conjugate heat equation) under the Ricci flow. We shall also derive Perelman's Harnack inequality for the fundamental solution of the conjugate heat equation under the Ricci flow.     

On the Positivity of the Stochastic Heat Equation

We study the positivity preserving properties of the heat equation with a white noise potential and random initial condition. Moreover, we find a generalized Feynman--Kac formula for the solution of the problem using methods from the white noise analysis. The initial condition can anticipate the driving white noise. We show that the solution is positive, when the random initial condition is positive. For the case of a time-dependent white noise potential, we give a special representation of the solution together with regularity results.     

Linear and nonlinear heat equations in $L^q_\delta$ spaces and universal bounds for global solutions

We develop a theory of both linear and nonlinear heat equations in the weighted Lebesgue spaces , where is the distance to the boundary. In particular, we prove an optimal estimate for the heat semigroup, and we establish sharp results on local existence-uniqueness and local nonexistence of solutions for semilinear heat equations with initial values in those spaces. This theory enables us to obtain new types of results concerning positive global solutions of superlinear parabolic problems. Namely, under certain assumptions, we prove that any global solution is uniformly bounded for by a universal constant, independent of the initial data. In all previous results, the bounds for global solutions were depending on the initial data. Received March 15, 2000 / Accepted October 18, 2000 / Published online February 5, 2001     

A finite‐time blow‐up result for a class of solutions with positive initial energy for coupled system of heat equations with memories

In this article, we are interested by a system of heat equations with initial condition and zero Dirichlet boundary conditions. We prove a finite‐time blow‐up result for a large class of solutions with positive initial energy.     

Time-Dependent Generalized Nash Equilibrium Problem

We prove an existence result for the time-dependent generalized Nash equilibrium problem under generalized convexity without neither a quasi-variational inequality reformulation nor a quasi-equilibrium problem reformulation. Furthermore, an application to the time-dependent abstract economy is considered.