Stability of Inverse Problems for Ultrahyperbolic Equations

In this paper, the authors consider inverse problems of determining a coefficient or a source term in an ultrahyperbolic equation by some lateral boundary data. The authors prove Hlder estimates which are global and local and the key tool is Carleman estimate.     

一类双曲型一维波动方程反问题解的整体唯一性

本文给出了一类双曲型一维波动方程反问题的解的整体唯一性定理。     

Stability Estimates for the Inverse Conductivity Problem for Less Regular Conductivities

A log-type stability estimate for the inverse conductivity problem in space dimension n ≥ 3, if the conductivity has C ^3/2+ε regularity is proven.     

Global Uniqueness for an Inverse Stochastic Hyperbolic Problem with Three Unknowns

This paper is addressed to an inverse stochastic hyperbolic problem with three unknowns, i.e., a random force intensity, an initial displacement, and an initial velocity. The global uniqueness for this inverse problem is proved by means of a new global Carleman estimate for the stochastic hyperbolic equation. It is found that both the formulation of stochastic inverse problems and the tools to solve them differ considerably from their deterministic counterpart. © 2015 Wiley Periodicals, Inc.     

一类双曲型方程反问题的存在性与唯一性

针对双曲型方程定解问题{utt=a2uxx+f(t),0xπ,a∈R且a≠0,u(0,t)=v1(t),u(π,t)=v2(t),t0,u(x,0)=g(x),ut(x,0)=h(x),0≤x≤π研究了可以唯一决定未知函数组{v1(t),v2(t),f(t)}的基本条件,提出了该定解问题的反问题,并且讨论了此反问题的存在性与唯一性.     

An Example of the Absence of a Global Solution to Some Inverse Problem for a Hyperbolic Equation

We give an example of an inverse problem for a hyperbolic equation which has a unique local solution but no global solutions.     

A Stability Estimate for a Solution to the Problem of Determination of Two Coefficients of a Hyperbolic Equation

We consider the problem of determination of two coefficients (x) and q(x) in a hyperbolic equation. Here (x) is the coefficient of the first derivative with respect to t and q(x) is the coefficient of the solution itself. We suppose that these coefficients are small in some norm and supported in a disk D. Oscillations are excited by the impulse function (t)(x·) supported on the straight line t=0, x·=0. Here is a unit vector playing the role of a parameter of the problem. The acoustic field generated by this source lying outside D is measured at the points of the boundary of D together with the normal derivative on some time interval of a fixed length T for two different values of the parameter . We prove that, for a sufficiently large T, the given information determines the sought coefficients uniquely. We obtain a stability estimate for a solution to the problem.     

一类双曲反问题的逼近算法及收敛性

该文考虑地球物理勘探中出现的间断特性阻抗的反演问题．利用样条插值理论，把无穷维空间上的反问题用有限维空间上的反问题来近似．利用半群理论，证明了近似反问题之解收敛于原反问题之解．据此可得到求解反问题的一种稳定的近似算法．     

Global Uniqueness and Stability for an Inverse Plate Problem

This paper deals with the global uniqueness and stability of an inverse problem for the plate equation. By means of a suitable transformation, the problem is reduced to the observability estimate for the plate equations with memory. The main tool that we employ is a global Carleman-type estimate. This work was supported by FANEDD of China Project 200119 and by NSFC under Grant 10371084. The author thanks Professor X. Zhang for help.     

Recovering of Damping Coefficients for a System of Coupled Wave Equations with Neumann Boundary Conditions: Uniqueness and Stability

The authors study the inverse problem of recovering damping coefficients for two coupled hyperbolic PDEs with Neumann boundary conditions by means of an additional measurement of Dirichlet boundary traces of the two solutions on a suitable, explicit subportion Γ1 of the boundary Γ, and over a computable time interval T > 0. Under sharp conditions on Γ0 = ΓnΓ1, T > 0, the uniqueness and stability of the damping coefficients are established. The proof uses critically the Carleman estimate due to Lasiecka et al. in 2000, together with a convenient tactical route “post-Carleman estimates” suggested by Isakov in 2006.     

A Stability Estimate for a Solution to a Two-Dimensional Inverse Problem of Electrodynamics

We consider the problem of finding the three coefficients c(x), (x), and q(x) in a hyperbolic equation. Here c(x) is the coefficient at the Laplace operator, (x) is the coefficient of the first time derivative, and q(x) is the coefficient of the lower-order term. The problem results from the inverse electrodynamic problem of finding the electrodynamic parameters of an isotropic medium under the assumption that the properties of the medium and the exterior current are independent of one coordinate. We suppose that the coefficients c(x)-1, (x), and q(x) are small in some norm and their supports are contained in some disk B. This is equivalent to the assumption that the electrodynamic parameters of the medium are close to constants. We suppose that the source initiating oscillations has the form of the impulse function (t)(x·) localized on the set t=0, x·=0. Here is a unit vector playing the role of a parameter of the problem. The electromagnetic field excited by this source applied outside B is measured at points of the boundary of the domain B on some time interval of a fixed length T counted from the moment of arrival of the signal from the source for three different values of the parameter . It is proven that, for a sufficiently large T, these data determine the sought coefficients uniquely. We obtain a conditional stability estimate for a solution to the problem.     

一类高阶非线性双曲型方程Cauchy问题解的爆破

本文研究一类高阶非线性双曲型方程utt-uxx+μuxxx-αuxxtt+βuxxxxtt=f(ux)x的Cauchy问题,证明问题解的存在性与唯一性,并给出解在有限时刻爆破的充分条件.     

一类非线性发展方程整体弱解的存在性和稳定性

该文考虑一类新的非线性方程(｜ut｜r-2ut)t-Δutt-Δu-ρ(t)Δut=f(u) 的初边值问题，利用小扰动法证明了整体弱解的 存在性，借用位势井的概念得到了解的稳定性.〖HT5”H〗关键词:〖HT5”SS〗非线性发展方程;初边值问题;整体弱解;稳定性.     

Stability for the Multi-Dimensional Borg-Levinson Theorem with Partial Spectral Data

We prove a stability estimate related to the multi-dimensional Borg-Levinson theorem of determining a potential from spectral data: the Dirichlet eigenvalues λ _k and the normal derivatives ?φ _k /?ν of the eigenfunctions on the boundary of a bounded domain. The estimate is of Hölder type, and we allow finitely many eigenvalues and normal derivatives to be unknown. We also show that if the spectral data is known asymptotically only, up to O(k ^?α) with α ? 1, then we still have Hölder stability.     

一个非标准逆热传导问题的具有对数稳定性的Fourier正则化方法

逆热传导问题是严重不适定问题,即问题的解(如果存在)将不连续依赖于测量数据,使其数值计算非常困难.但最近20 年来人们主要关注标准的逆热传导问题而对非标准情形研究较少.本文给出了一个非标准逆热传导问题的具有对数稳定性的Fourier正则化方法,恢复了解对数据的连续依赖性.     

Local Stability for an Inverse Coefficient Problem of a Fractional Diffusion Equation

Time-fractional diffusion equations are of great interest and importance on describing the power law decay for diffusion in porous media. In this paper, to identify the diffusion rate, i.e., the heterogeneity of medium, the authors consider an inverse coefficient problem utilizing finite measurements and obtain a local Hlder type conditional stability based upon two Carleman estimates for the corresponding differential equations of integer orders.     

实对称矩阵的一类逆特征值问题

利用矩阵的奇异值分解及广义逆，给出了矩阵约束下矩阵反问题AX=B有实对称解的充分必要条件及其通解的表达式．此外，给出了在矩阵方程的解集合中与给定矩阵的最佳逼近解的表达式．     

一类Laplace双曲型方程广义Cauchy问题的反问题

本文讨论了确定Ｌａｐｌｉｔｃｅ双曲型方程u_xy(x,y)+a(x,y)u_x(x,y)+b(x,y)+u_y(x,y)+q(x)u(x,y)=f(x,y)的广义Ｃａｕｃｈｙ问题中系数ｑ（ｘ）的反问题。文中利用特征法线及不动点理论，导出了与反问题等价的非线性积分方程组，证明了反问题局部解的存在唯一性，最后给出了反问题整体用的唯一性定理。     

Uniform Stability Estimates for Constant-Coefficient Symmetric Hyperbolic Boundary Value Problems

Answering a question left open in Métivier and Zumbrun (2005 Métivier , G. , Zumbrun , K. ( 2005 ). Variable multiplicities, hyperbolic boundary value problems for symmetric systems with variable multiplicities . J. Diff. Eq. 211 ( 1 ): 61 – 134 . [Google Scholar]), we show for general symmetric hyperbolic boundary problems with constant coefficients, including in particular systems with characteristics of variable multiplicity, that the uniform Lopatinski condition implies strong L ² well-posedness, with no further structural assumptions. The result applies, more generally, to any system that is strongly L ² well-posed for at least one boundary condition. The proof is completely elementary, avoiding reference to Kreiss symmetrizers or other specific techniques. On the other hand, it is specific to the constant-coefficient case; at least, it does not translate in an obvious way to the variable-coefficient case. The result in the hyperbolic case is derived from a more general principle that can be applied, for example, to parabolic or partially parabolic problems like the Navier–Stokes or viscous MHD equations linearized about a constant state or even a viscous shock.     

Global Existence and Uniqueness of Solution for a Class of Non-Newtonian Fluids

In this paper, we consider a class of non-Newtonian fluids in one-dimensional bounded interval. The global existence and uniqueness of solution are investigated.