抛物型方程的一个反问题

A Reconstruction Algorithm for a Semilinear Parabolic Inverse Problem

In this paper, we consider an inverse problem to determine a semilinear term of a parabolic equation from a single boundary measurement of Neumann type. For this problem, a reconstruction algorithm is established by the spectral representation for the fundamental solution of heat equation with homogeneous Neumann boundary condition and the Whitney extension theorem.

Lipschitz stability for an inverse source problem in anisotropic parabolic equations with dynamic boundary conditions

In this paper, we study an inverse problem for linear parabolic system with variable diffusion coefficients subject to dynamic boundary conditions. We prove a global Lipschitz stability for the inverse problem involving a simultaneous recovery of two source terms from a single measurement and interior observations, based on a recent Carleman estimate for such problems.

Inverse moving source problems for parabolic equations

This paper is concerned with the inverse moving source problems for parabolic equations. Given the temporal function, we prove the uniqueness of the nonlinear inverse problem of determining the orbit function by final data measured in a bounded domain. On the other hand, given the orbit function we also show that the profile function can be uniquely determined by final data measured in a bounded domain away from the domain enclosing the moving orbit. The proofs adopt the Fourier approach and results from complex analysis.

Inverse parabolic problems by Carleman estimates with data taken initial or final time moment of observation

We consider a parabolic equation in a bounded domain $\OOO$ over a time interval $(0,T)$ with the homogeneous Neumann boundary condition. We arbitrarily choose a subboundary $Γ\subset \ppp\OOO$. Then, we discuss an inverse problem of determining a zeroth-order spatially varying coefficient by extra data of solution $u$: $u\vert_{Γ\times (0,T)}$ and $u(\cdot,t_0)$ in $\OOO$ with $t_0=0$ or $t=T$. First we establish a conditional Lipschitz stability estimate as well as the uniqueness for the case $t_0=T.$ Second, under additional condition for $Γ$, we prove the uniqueness for the case $t_0=0$. T…

一种抛物方程反源问题的数值方法及其在系数反问题中的应用

A numerical method for an inverse source problem for parabolic equations and its application to a coefficient inverse problem

本文提出两阶段数值方法求解抛物方程反源问题，先建立解的时间导数方程再用拟可逆方法求解，并将该反源问题作为非线性系数反问题的线性化形式迭代求解，给出了两类问题的数值结果。

一类抛物型方程的单观测Carlemen估计的逆传导问题

Inverse Conductivity Problem for a Parabolic Equation using a Carlemen Estimate with one Observation

对于有界区域上的热方程，我们给出了一个光滑扩散系数的稳定性结果。关键成分是一个整体的Carleman型估计，庞加莱型估计和能量估计与一个单一的观察作用在边界的一部分。

Inverse problem for recovery of temporal component of source term for multi-term time fractional parabolic equation with nonlocal boundary datum

Inverse problem for multi-term fractional parabolic equation in two dimensional space, involving m + 1 Caputo fractional derivatives in time, is investigated. Presence of nonlocal boundary conditions leads to a non-self-adjoint spectral problem. A bi-orthogonal system of functions is used to construct the solution that involves double infinite series. Properties of multinomial Mittag-Leffler function and eigenfunctions are used to prove the classical nature of the solution under certain regularity conditions on the given datum.

A parabolic inverse problem with mixed boundary data. Stability estimates for the unknown boundary and impedance

We consider the problem of determining an unaccessible part of the boundary of a conductor by mean of thermal measurements. We study a problem of corrosion where a Robin type condition is prescribed on the damaged part and we prove logarithmic stability estimate.

反随机源及Cauchy问题研究：半离散随机抛物方程在任意维度中的逆问题

Inverse Random Source and Cauchy Problems for Semi-Discrete Stochastic Parabolic Equations in Arbitrary Dimensions

本文研究了半离散随机抛物方程在任意维度中的逆源和Cauchy逆问题，利用新的Carleman估计获得稳定性结果。

抛物型方程反问题2

Inverse problems for parabolic equations 2

令$0\leq x \leq π, t\geq 0.$中的$u_t-u_{xx}=h(t)$假设$u(0,t)=v(t)$、$u(π,t)=0$和$u(x,0)=g(t)$。问题是：{ \it什么额外的数据唯一地确定三个未知函数$\{h, v, g\}$？}。回答了这个问题，并提出了上述三个功能的恢复的分析方法。

Convergence Rates of Tikhonov Regularizations for Elliptic and Parabolic Inverse Radiativity Problems

We shall study in this paper the Lipschitz type stabilities and convergence rates of Tikhonov regularization for the recovery of the radiativities in elliptic and parabolic systems with Dirichlet boundary conditions. The Lipschitz type stability estimates are derived. Due to the difficulty of the verification of the existing source conditions or nonlinearity conditions for the considered inverse radiativity problems in high dimensional spaces, some new variational source conditions are proposed. The conditions are rigorously verified in general dimensional spaces under the Lipschitz type stabi…

Well-posedness and inverse Robin estimate for a multiscale elliptic/parabolic system

We establish the well-posedness of a coupled micro-macro parabolic-elliptic system modeling the interplay between two pressures in a gas-liquid mixture close to equilibrium that is filling a porous media with distributed microstructures. Additionally, we prove a local stability estimate for the inverse micro-macro Robin problem, potentially useful in identifying quantitatively a micro-macro interfacial Robin transfer coefficient given microscopic measurements on accessible fixed interfaces. To tackle the solvability issue we use two-scale energy estimates and two-scale regularity/compactness a…

Inverse source problem and null controllability for multidimensional parabolic operators of Grushin type

The approach to Lipschitz stability for uniformly parabolic equations introduced by Imanuvilov and Yamamoto in 1998, based on Carleman estimates, seems hard to apply to the case of Grushin-type operators of interest to this paper. Indeed, such estimates are still missing for parabolic operators degenerating in the interior of the space domain. Nevertheless, we are able to prove Lipschitz stability results for inverse source problems for such operators, with locally distributed measurements in arbitrary space dimension. For this purpose, we follow a mixed strategy which combines the appraoch du…

度量星图上带积分超定条件的时空分数阶抛物型方程逆源问题

Inverse source problem for the space-time fractional parabolic equation on a metric star graph with an integral overdetermination condition

本文研究度量星图上分数阶抛物型方程初边值问题，基于先验估计证明强解存在唯一性，并首次通过算子方程转化研究带积分超定条件的时空分数阶逆源问题，证明预解算子适定。

Inverse source problems for time-fractional mixed parabolic-hyperbolic type equations

In the present paper we consider an inverse source problem for time-fractional mixed parabolic-hyperbolic equation with the Caputo derivative. In case, when hyperbolic part of the considered mixed type equation is wave equation, the uniqueness of source and solution are strongly influenced by initial time and generally is ill-posed. However, when the hyperbolic part is time fractional, the problem is well posed if end time is large. Our method relies on the orthonormal system of eigenfunctions of the operator with respect to space variable. We proved the uniqueness and stability of certain wea…

The Calderón problem for nonlocal parabolic operators

We investigate inverse problems in the determination of leading coefficients for nonlocal parabolic operators, by knowing the corresponding Cauchy data in the exterior space-time domain. The key contribution is that we reduce nonlocal parabolic inverse problems to the corresponding local inverse problems with the lateral boundary Cauchy data. In addition, we derive a new equation and offer a novel proof of the unique continuation property for this new equation. We also build both uniqueness and non-uniqueness results for both nonlocal isotropic and anisotropic parabolic Calderón problems, resp…

半离散抛物算子反问题中的Lipschitz稳定性与重构

Lipschitz stability and reconstruction in inverse problems for semi-discrete parabolic operators

针对半离散抛物方程的反问题，本文利用受网格尺寸约束的新颖半离散Carleman估计，建立了恢复势函数的稳定性估计并提出了收敛的重构算法。

基于单次测量的含初值抛物型反问题

Inverse parabolic problem with initial data by a single measurement

该研究针对带齐次Neumann边界条件的抛物型初边值问题，通过约化为椭圆型反问题并结合椭圆Carleman估计，证明了利用子边界上单次Dirichlet测量确定零阶项空间变系数的唯一性，且唯一性在初值为正的光滑子域内成立。

抛物方程中点源逆问题的稳定性估计

Stability Estimates for the Inverse Problem of Reconstructing Point sources in Parabolic Equations

本文提出了一种结合多种方法的新型方法，用于推导抛物方程中点源逆问题的稳定性估计，并通过数值重建验证了理论结果。

由一个分量测量的双分量抛物型方程组的反问题

Inverse problem for a parabolic system with two components by measurements of one component

考虑一类具有一阶和零阶耦合的抛物型方程的$2\times 2$系统，在没有初值数据的情况下，仅用一个分量的额外数据建立了Carleman估计.然后，我们应用Carleman估计确定的一些或所有的系数，在一个任意的子域中的观察在一个时间间隔内只有一个组件和数据的两个组件在一个固定的正时间$θ$在整个空间域的反问题。主要结果是反问题的Lipschitz稳定性估计。对于Lipschitz稳定性，我们必须假设两个分量在$θ$处的一些非退化条件，并且对于它，我们可以近似地通过仅输入到一个分量来控制$2 \times 2$系统的两个分量。这种近似可控性也证明了我们的新Carleman估计。最后，我们建立了零阶项与一个分量耦合的抛物型方程的$3\times 3$系统的Carleman估计，证明了相应的近似能控性。

带无界零阶系数的方程的两个抛物型反问题

Two parabolic inverse problems for an equation with unbounded zero-order coefficient

该研究针对带无界零阶系数的抛物型方程，证明了边界测量下系数确定的唯一性（覆盖最大类无界零阶系数），建立了单次内部测量下初值确定的对数稳定性不等式，并借助带无界势的Schrödinger方程的新全局定量唯一延拓性得到对应能观性不等式。

Stability of the determination of a time-dependent coefficient in parabolic equations

We establish a Lipschitz stability estimate for the inverse problem consisting in the determination of the coefficient $σ(t)$, appearing in a Dirichlet initial-boundary value problem for the parabolic equation $\partial_tu-Δ_x u+σ(t)f(x)u=0$, from Neumann boundary data. We extend this result to the same inverse problem when the previous linear parabolic equation in changed to the semi-linear parabolic equation $\partial_tu-Δ_x u=F(t,x,σ(t),u(x,t))$.

弱耦合抛物系统反系数问题的唯一性

Uniqueness for an inverse coefficient problem of a weakly coupled parabolic system

本文研究了弱耦合抛物系统在边界观测下的唯一性问题，通过特征函数展开和Gel'fand-Levitan理论扩展证明了系数矩阵P(x)的唯一性。

Inverse problems for parabolic equations 3

Let $u_t-a(t)u_{xx}=f(x, t)$ in $0\leq x \leq π,\,\,t\geq 0.$ Assume that $u(0,t)=u_1(t)$, $u(π,t)=u_2(t)$, $u(x,0)=h(x)$, and the extra data $u_x(0,t)=g(t)$ are known. The inverse problem is: {\it How does one determine the unknown $a(t)$?} The function $a(t)>a_0>0$ is assumed continuous and bounded. This question is answered and a method for recovery of $a(t)$ is proposed. There are several papers in which sufficient conditions are given for the uniqueness and existence of $a(t)$, but apparently there was no method proposed for calculating of $a$. The method given in this paper for proving t…

抛物方程在逆平方无限势阱中的可控性 via 全局Carleman估计

Controllability of parabolic equations with inverse square infinite potential wells via global Carleman estimates

本文通过全局Carleman估计，首次在多维空间中解决了抛物方程在逆平方无限势阱中的可控性问题。

带动态边界条件的抛物型系统的可控性与反问题

Controllability and Inverse Problems for Parabolic Systems with Dynamic Boundary Conditions

本文是一篇综述，梳理带动态边界条件的抛物型系统零可控性与反问题的研究成果，阐释Carleman估计等经典方法的推广应用，分析其相较静态边界条件的难点，并给出结论与开放问题。

Classification of local asymptotics for solutions to heat equations with inverse-square potentials

Asymptotic behavior of solutions to heat equations with spatially singular inverse-square potentials is studied. By combining a parabolic Almgren type monotonicity formula with blow-up methods, we evaluate the exact behavior near the singularity of solutions to linear and subcritical semilinear parabolic equations with Hardy type potentials. As a remarkable byproduct, a unique continuation property is obtained.

通过非负解的边界观测确定抛物型系统及其生物学应用

Determining a parabolic system by boundary observation of its non-negative solutions with biological applications

本文通过边界观测非负解，利用高阶变分方案创新逐次线性化方法，确定耦合非线性抛物型系统系数，并应用于生态种群模型反问题。

从部分边界数据确定抛物型方程中非线性项的稳定性与重构

Stability and Reconstruction of a Nonlinearity in a Parabolic Equation from Partial Boundary Data

研究从边界任意子集的单次通量确定非线性抛物型方程半线性项的反问题，建立赫尔德型稳定性估计，提出迭代重构算法并通过数值实验验证其准确性。

一般抛物型方程反问题的正则性

Regularity of a inverse problem for generic parabolic equations

研究一类最简单的抛物型方程的反边值问题，其中齐次Cauchy条件在初始时刻是不适定的。对于一类广泛的边值输入，建立了解的一些正则性。

Inverse coefficient problem for Grushin-type parabolic operators

The approach to Lipschitz stability for uniformly parabolic equations introduced by Imanuvilov and Yamamoto in 1998 based on Carleman estimates, seems hard to apply to the case of Grushin-type operators studied in this paper. Indeed, such estimates are still missing for parabolic operators degenerating in the interior of the space domain. Nevertheless, we are able to prove Lipschitz stability results for inverse coefficient problems for such operators, with locally distributed measurements in arbitrary space dimension. For this purpose, we follow a strategy that combines Fourier decomposition …

Corrigendum: Recover the source and initial value simultaneously in a parabolic equation

In this note, the corrigendum to our earlier paper "Recover the source and initial value simultaneously in a parabolic equation" [2014 Inverse Problems 30 065013] is given.

Weighted Hardy inequalities and Ornstein-Uhlenbeck type operators perturbed by multipolar inverse square potentials

We give necessary and sufficient conditions for the existence of weak solutions of a parabolic problem corresponding to the Kolmogorov operators perturbed by a multipolar inverse square potential with respect to the Gaussian probability measure which is the unique invariant measure for Ornstein-Uhlenbeck type operators. We state the optimality of the constant and, then, the nonexistence of positive exponentially bounded solutions to the parabolic problem.

Direct and Inverse problems for time-fractional pseudo-parabolic equations

The purpose of this paper is to establish the solvability results to direct and inverse problems for time-fractional pseudo-parabolic equations with the self-adjoint operators. We are especially interested in proving existence and uniqueness of the solutions in the abstract setting of Hilbert spaces.

非局部抛物算子的纠缠原理与分数阶Calderón问题

Entanglement principle and fractional Calderón problem for nonlocal parabolic operators

本文针对变系数非局部抛物算子的反问题，提出全新纠缠原理并证明可通过狄利克雷-诺伊曼映射唯一确定低阶扰动，还推导修正版纠缠原理解决直接应用时解正则性不足的问题。

An inverse problem of radiative potentials and initial temperatures in parabolic equations with dynamic boundary conditions

We study an inverse problem involving the restoration of two radiative potentials, not necessarily smooth, simultaneously with initial temperatures in parabolic equations with dynamic boundary conditions. We prove a Lipschitz stability estimate for the relevant potentials using a recent Carleman estimate, and a logarithmic stability result for the initial temperatures by a logarithmic convexity method, based on observations in an arbitrary subdomain.

时间分数伪抛物型方程的正反问题研究

Forward and inverse problems for a time-fractional pseudo-parabolic equation with variable coefficients

本文研究了时间分数伪抛物型方程的正反问题，针对时间依赖系数σ(t)进行推广，采用傅里叶方法证明解的存在唯一性，并提出数值算法。逆问题通过过定条件确定时间依赖源函数，证明了解的存在性和唯一性。

Inverse problems for general parabolic systems and application to Ornstein-Uhlenbeck equation

We investigate the link between inverse problems and final state observability for a general class of parabolic systems. We generalize a stability result for initial data due to García and Takahashi [16], known for the case of self-adjoint dissipative operators. More precisely, we consider a system governed by an analytic semigroup. Under the assumption of final state observability, we prove a logarithmic stability estimate depending on the analyticity angle of the semigroup. This is done by using a general logarithmic convexity result. The abstract result is illustrated by considering the Orn…

某些双非线性抛物型和椭圆型方程的逆边界值问题

Inverse boundary value problems for certain doubly nonlinear parabolic and elliptic equations

本文研究了双非线性抛物型和椭圆型方程的逆边界值问题，通过将抛物型问题转化为椭圆型问题，证明了在特定条件下系数的唯一性。

反演源问题对于抛物方程的稀疏移动观测

Inverse source problem for the parabolic equation with sparse moving observations

本文研究了通过稀疏边界测量反演抛物方程源项的问题，提出了一种基于移动传感器的数据定位方法，并通过数值实验验证了算法的有效性。

具奇异位势的高阶抛物型方程的Baras-Goldstein型解的不存在性

On nonexistence of Baras--Goldstein type for higher-order parabolic equations with singular potentials

将Baras和Goldstein（1984）关于具有逆奇异势的热传导方程的不存在性结果推广到具有Hardy-超临界奇异势的2 m阶线性抛物型方程.扩展到其他线性和非线性奇异偏微分方程进行了讨论。

An inverse problem of identifying the radiative coefficient in a degenerate parabolic equation

This work investigates an inverse problem of determining the radiative coefficient in a degenerate parabolic equation from the final overspecified data. Being different from other inverse coefficient problems in which the principle coefficients are assumed to be strictly positive definite, the mathematical model discussed in the paper belongs to the second order parabolic equations with non-negative characteristic form, namely that there exists degeneracy on the lateral boundaries of the domain. The uniqueness of the solution is obtained by the contraction mapping principle. Based on the optim…

确定未知时变边界的一类随机抛物型反问题的稳定性估计

Stability Estimate for an Inverse Stochastic Parabolic Problem of Determining Unknown Time-varying Boundary

本文研究随机抛物型方程中未知时变边界的反问题，利用新的Carleman估计证明未知边界对数依赖于内部测量，从而可从任意内部区域观测跟踪边界行为。

Strong maximum principle for fractional diffusion equations and an application to an inverse source problem

The strong maximum principle is a remarkable characterization of parabolic equations, which is expected to be partly inherited by fractional diffusion equations. Based on the corresponding weak maximum principle, in this paper we establish a strong maximum principle for time-fractional diffusion equations with Caputo derivatives, which is slightly weaker than that for the parabolic case. As a direct application, we give a uniqueness result for a related inverse source problem on the determination of the temporal component of the inhomogeneous term.

Inverse problem of reconstruction of degenerate diffusion coefficient in a parabolic equation

We consider the inverse problem of identification of degenerate diffusion coefficient of the form $x^αa(x)$ in a one dimensional parabolic equation by some extra data. We first prove by energy methods the uniqueness and Lipschitz stability results for the identification of a constant coefficient $a$ and the power $α$ by knowing an interior data at some time. On the other hand, we obtain the uniqueness result for the identification of a general diffusion coefficients $a(x)$ and also the power $α$ form a boundary data on one side of the space interval. The proof is based on global Carleman estim…

关于具有容量密度条件的领域上抛物型方程的全局Hölder可解性

Global Hölder Solvability of parabolic equations on domains with capacity density conditions

本文研究了具有容量密度条件的领域上抛物型方程的全局Hölder解的存在性，允许数据在边界附近具有临界奇异性。

抛物方程退化导电区域的重构

Reconstruction of degenerate conductivity region for parabolic equations

该研究通过边界法向导数测量重构一维抛物方程扩散系数的退化点，分析了初始数据对唯一性和稳定性的影响，并利用贝塞尔函数显式解及数值实验验证了理论结果。

Estimates of solutions for the parabolic $p$-Laplacian equation with measure via parabolic nonlinear potentials

For weak solutions to the evolutional $p$-Laplace equation with a time-dependent Radon measure on the right hand side we obtain pointwise estimates via a nonlinear parabolic potential.

Carleman commutator approach in logarithmic convexity for parabolic equations

In this paper we investigate on a new strategy combining the logarithmic convexity (or frequency function) and the Carleman commutator to obtain an observation estimate at one time for the heat equation in a bounded domain. We also consider the heat equation with an inverse square potential. Moreover, a spectral inequality for the associated eigenvalue problem is derived.

确定半线性抛物型偏微分方程源项的条件稳定性

Conditional stability in determining source terms of semilinear parabolic partial differential equations

本文提出了一种非线性反源问题的条件Hölder稳定性方法，结合广义微分算子线性化和Carleman估计，用于确定半线性抛物型方程的源项。

一维随机抛物方程空间半离散逼近的Carleman估计及其应用

Carleman estimates for space semi-discrete approximations of one-dimensional stochastic parabolic equation and its applications

本文为一维随机抛物方程的空间半离散格式建立齐次与非齐次边界下的Carleman估计，并分别用于证明离散反随机源问题的Lipschitz稳定性和离散Cauchy问题的Hölder稳定性。

The parabolic flows for complex quotient equations

We apply the parabolic flow method to solving complex quotient equations on closed Kähler manifolds. We study the parabolic equation and prove the convergence. As a result, we solve the complex quotient equations.

Weak unique continuation property and a related inverse source problem for time-fractional diffusion-advection equations

In this paper, we first establish a weak unique continuation property for time-fractional diffusion-advection equations. The proof is mainly based on the Laplace transform and the unique continuation properties for elliptic and parabolic equations. The result is weaker than its parabolic counterpart in the sense that we additionally impose the homogeneous boundary condition. As a direct application, we prove the uniqueness for an inverse problem on determining the spatial component in the source term in by interior measurements. Numerically, we reformulate our inverse source problem as an opti…

Hopf's lemmas for parabolic fractional Laplacians and parabolic fractional $p$-Laplacians

In this paper, we first establish Hopf's lemmas for parabolic fractional equations and parabolic fractional $p$-equations. Then we derive an asymptotic Hopf's lemma for antisymmetric solutions to parabolic fractional equations. We believe that these Hopf's lemmas will become powerful tools in obtaining qualitative properties of solutions for nonlocal parabolic equations.