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。
英文摘要：
Let $u_t-u_{xx}=h(t)$ in $0\leq x \leq π, t\geq 0.$ Assume that $u(0,t)=v(t)$, $u(π,t)=0$, and $u(x,0)=g(t)$. The problem is: {\it what extra data determine the three unknown functions $\{h, v, g\}$ uniquely?}. This question is answered and an analytical method for recovery of the above three functions is proposed.
In this paper we investigate direct and inverse problems for time-fractional pseudo-parabolic equations associated with the Jacobi operator. The existence and uniqueness of the solutions are proved. Also, the stability result of the inverse source problem (ISP) is established.
In this work, we shall study the nonlinear inverse problems of recovering the Robin coefficients in elliptic and parabolic systems of second order, and establish their local Lipschitz stabilities. Some local Lipschitz stability was derived for an elliptic inverse Robin problem. We shall first restructure the arguments in \cite{chou04} for the local Lipschitz stability so that the stability follows from three basic conditions for the elliptic inverse Robin problem. The new arguments are then generalized to help establish a novel local Lipschitz stability for parabolic inverse Robin problems.
We study a class of fractional parabolic equations involving a time-dependent magnetic potential and formulate the corresponding inverse problem. We determine both the magnetic potential and the electric potential from the exterior partial measurements of the Dirichlet-to-Neumann map.
In this article, We investigate an inverse problem of determining the time-dependent source factor in parabolic integro-differential equations from boundary data. We establish the uniqueness and the conditional stability estimate of Hölder type for the inverse source problem in a cylindrical shaped domain. Our methodology is based on the Bukhgeim-Klibanov method by means of the Carleman estimate. Here we also derive the "time-like" Carleman estimate for parabolic integro-differential equations.
In this article, for a fourth-order parabolic equation which is closely related for example to the Cahn-Hilliard equation, we study an inverse source problem by interior data and the continuation of solution from lateral Cauchy data. Our method relies on a Carleman estimate and proves conditional stability for both problems.
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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.
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.
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.
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…
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.
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.
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…
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…
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…
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…
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…
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 $\partial_tu-Δ_x u=F(t,x,σ(t),u(x,t))$.
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…
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.
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 …
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.
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.
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.
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.
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…
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…
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.
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…
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.
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.
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.
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…
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.
