Peter Hintz, Gunther Uhlmann, Jian Zhai
We consider an inverse boundary value problem for a semilinear wave equation on a time-dependent Lorentzian manifold with time-like boundary. The time-dependent coefficients of the nonlinear terms can be recovered in the interior from the knowledge of the Neumann-to-Dirichlet map. Either distorted plane waves or Gaussian beams can be used to derive uniqueness.
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Unique inversion of smooth nonlinearity in semilinear wave equations on Lorentzian manifolds
研究洛伦兹流形上半线性波动方程光滑非线性项的恢复，核心方法是结合高阶线性化与扭曲平面波、高斯光束，利用多波相互作用恢复泰勒系数。
We consider the semilinear wave equation $\Box_g u+a u^4=0$, $a\neq 0$, on a Lorentzian manifold $(M,g)$ with timelike boundary. We show that from the knowledge of the Dirichlet-to-Neumann map one can recover the metric $g$ and the coefficient $a$ up to natural obstructions. Our proof rests on the analysis of the interaction of distorted plane waves together with a scattering control argument, as well as Gaussian beam solutions.
We consider inverse problems in space-time $(M, g)$, a $4$-dimensional Lorentzian manifold. For semilinear wave equations $\square_g u + H(x, u) = f$, where $\square_g$ denotes the usual Laplace-Beltrami operator, we prove that the source-to-solution map $L: f \rightarrow u|_V$, where $V$ is a neighborhood of a time-like geodesic $μ$, determines the topological, differentiable structure and the conformal class of the metric of the space-time in the maximal set where waves can propagate from $μ$ and return back. Moreover, on a given space-time $(M, g)$, the source-to-solution map determines som…
For semilinear wave equations on Lorentzian manifolds with quadratic derivative non-linear terms, we study the inverse problem of determining the background Lorentzian metric. Under some conditions on the nonlinear term, we show that from the source-to-solution map, one can determine the Lorentzian metric up to diffeomorphisms.
We show that the scattering operator for defocusing energy critical semilinear wave equations \square u+f(u)=0, with f C-infinity and f ~ u^5, in three space dimensions, determines f.
We show that the knowledge of the Dirichlet-to-Neumann map on an arbitrary open portion of the boundary of a domain in $\mathbb{R}^n$, $n\ge 2$, for a class of semilinear elliptic equations, determines the nonlinearity uniquely.
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.
An inverse problem to determine a space-dependent factor in a semilinear time-fractional diffusion equation is considered. Additional data are given in the form of an integral with the Borel measure over the time. Uniqueness of the solution of the inverse problem is studied. The method uses a positivity principle of the corresponding differential equation that is also proved in the paper.
We establish a logarithmic stability inequality for the inverse problem of determining the non linear term, appearing in a semilinear BVP, from the corresponding Dirichlet-to-Neumann map (abbreviated to DtN map in the rest of this text). Our result can be seen as a stability inequality for an earlier uniqueness result by Isakov and Sylvester [Commun. Pure Appl. Math. 47 (1994), 1403-1410].
We study a class of fractional semilinear elliptic equations and formulate the corresponding Calderón problem. We determine the nonlinearity from the exterior partial measurements of the Dirichlet-to-Neumann map by using first order linearization and the Runge approximation property.
We show that the linear span of the set of scalar products of gradients of harmonic functions on a bounded smooth domain $Ω\subset \mathbb{R}^n$ which vanish on a closed proper subset of the boundary is dense in $L^1(Ω)$. We apply this density result to solve some partial data inverse boundary problems for a class of semilinear elliptic PDE with quadratic gradient terms.
We study the direct and inverse scattering problem for the semilinear Schrödinger equation $Δu+a(x,u)+k^2u=0$ in $\mathbb{R}^d$. We show well-posedness in the direct problem for small solutions based on the Banach fixed point theorem, and the solution has the certain asymptotic behavior at infinity. We also show the inverse problem that the semilinear function $a(x,z)$ is uniquely determined from the scattering data. The idea is certain linearization that by using sources with several parameters we differentiate the nonlinear equation with respect to these parameter in order to get the linear …
An inverse problem for semilinear wave equations on metric tree graphs
研究度量树图上半线性波动方程的反问题，通过Dirichlet到Neumann映射恢复图的连接性、边长、时间不变势场和非线性项的时间依赖系数。
An inverse problem for semilinear equations involving the fractional Laplacian
本文研究带零阶非线性扰动的分数阶拉普拉斯热方程与波动方程反问题，利用分数阶 Dirichlet-to-Neumann 映射、Runge 逼近和唯一延拓性质恢复非线性项。
We consider the inverse problem of determining a potential in a semilinear elliptic equation from the knowledge of the Dirichlet-to-Neumann map. For bounded Euclidean domains we prove that the potential is uniquely determined by the Dirichlet-to-Neumann map measured at a single boundary point, or integrated against a fixed measure. This result is valid even when the Dirichlet data is only given on a small subset of the boundary. We also give related uniqueness results on Riemannian manifolds.
In this paper, we establish blow-up results for the semilinear wave equation in generalized Einstein-de Sitter spacetime with nonlinearity of derivative type. Our approach is based on the integral representation formula for the solution to the corresponding linear problem in the one-dimensional case, that we will determine through Yagdjian's Integral Transform approach. As upper bound for the exponent of the nonlinear term, we discover a Glassey-type exponent which depends both on the space dimension and on the Lorentzian metric in the generalized Einstein-de Sitter spacetime.
Hölder stability for a semilinear elliptic inverse problem
本研究针对半线性椭圆方程非线性项的反演问题，改进了已有对数型稳定性结果，在更少边界测量和更低正则性条件下得到Hölder稳定性估计，采用构造子边界零值特殊解的方法完成证明。
Inverse problems for semilinear elliptic equations with low regularity
研究在固定解邻域内通过边界测量唯一确定非线性项，方法涉及低正则性逆问题理论，贡献为扩展了近期研究成果。
This paper is concerned with the forward and inverse problems for the fractional semilinear elliptic equation $(-Δ)^s u +a(x,u)=0$ for $0<s<1$. For the forward problem, we proved the problem is well-posed and has a unique solution for small exterior data. The inverse problems we consider here consists of two cases. First we demonstrate that an unknown coefficient $a(x,u)$ can be uniquely determined from the knowledge of exterior measurements, known as the Dirichlet-to-Neumann map. Second, despite the presence of an unknown obstacle in the media, we show that the obstacle and the coefficient ca…
Forward and inverse problems of a semilinear transport equation
研究半线性辐射传输模型的正反问题，改进了边界数据理论，并统一了扩散和传输 regime 的L1稳定性理论。
Well-posedness and inverse problems for semilinear nonlocal wave equations
本文研究半线性非局部波动方程，建立适定性，并利用外部Dirichlet-to-Neumann映射和被动测量恢复非线性项及初始数据，结果适用于任意维数。
In this paper, we study the semilinear wave equations with the inverse-square potential. By transferring the original equation to a "fractional dimensional" wave equation and analyzing the properties of its fundamental solution, we establish a long-time existence result, for sufficiently small, spherically symmetric initial data. Together with the previously known blow-up result, we determine the critical exponent which divides the global existence and finite time blow-up. Moreover, the sharp lower bounds of the lifespan are obtained, except for certain borderline case. In addition, our techno…
We study dynamics of interfaces in solutions of the equation $ε\Box u + \frac 1 εf_ε(u)=0$, for $f_ε$ of the form $f_ε(u) = (u^2-1)(2u- εκ)$, for $κ\in {\mathbb R}$, as well as more general, but qualitatively similar, nonlinearities. We consider equations of this form both in $(1+n)$-dimensional Minkowski space, $n\ge 1$, and on certain more general Lorentzian manifolds, and we prove that for suitable initial data, solutions exhibit interfaces that sweep out timelike hypersurfaces of mean curvature proportional to $κ$. In particular, in 1 dimension these interfaces behave like a relativistic p…
Inverse problem for connections in semi-linear wave equations on Lorentzian manifolds
针对任意全局双曲洛伦兹流形上带三次非线性的半线性波动方程，本文利用非线性波相互作用的微局部分析及断裂光线变换反演，恢复其 Hermitian 联络。
An Inverse Almost Periodic Problem for a Semilinear Strongly Damped Wave Equation
本文研究了带有Dirichlet边界条件的强阻尼半线性波动方程的反边界值问题，通过积分过定条件确定时间依赖源系数，建立有限时间区间上的解的存在性和唯一性，并将其扩展到半轴上，构造整体实数轴上的有界强解。
We study inverse source problems associated to semilinear elliptic equations of the form \[ Δu(x)+a(x,u)=F(x), \] on a bounded domain $Ω\subset \mathbb{R}^n$, $n\geq 2$. We show that it is possible to use nonlinearity to break the gauge symmetry of the inverse source problem for a class of nonlinearities $a(x,u)$. This is in contrast to inverse source problems for linear equations, which always have a gauge symmetry. The class of nonlinearities include certain polynomials and exponential nonlinearities. For these nonlinearities, we determine both $a(x,u)$ and $F(x)$ uniquely from the associate…
An inverse problem for semilinear elliptic equations with generalized Kerr-type nonlinearities
本文提出了一种扩展的包围法，用于解决具有广义 Kerr 型非线性项的半线性椭圆方程的反问题，适用于多种非线性模型。
A partial data coefficient identification inverse problem for a semilinear damped wave operator
针对ℝ^(1+d)（d≥2）有界域中的半线性阻尼波动算子系数识别反问题，研究人员利用唯一延拓原理等方法，从部分Dirichlet-to-Neumann映射实现了三类系数的唯一恢复。
We study various partial data inverse boundary value problems for the semilinear elliptic equation $Δu+ a(x,u)=0$ in a domain in $\mathbb R^n$ by using the higher order linearization technique introduced in [LLS 19, FO19]. We show that the Dirichlet-to-Neumann map of the above equation determines the Taylor series of $a(x,z)$ at $z=0$ under general assumptions on $a(x,z)$. The determination of the Taylor series can be done in parallel with the detection of an unknown cavity inside the domain or an unknown part of the boundary of the domain. The method relies on the solution of the linearized p…
Inverse Nonlinear Scattering by a Metric
本文研究从半线性波动方程的散射算子反推含时全局双曲洛伦兹度规的反问题。
An inverse coefficient problem for a semilinear wave equation by the first order linearization
研究半线性波动方程中未知系数的恢复问题，核心方法是结合一阶线性化与容斥原理恒等式，利用波动方程几何光学解，还通过数值示例表明基于神经网络的反演算法可有效重建未知系数。
In this paper we study nonexistence of non-negative distributional supersolutions for a class of semilinear elliptic equations involving inverse-square potentials.
Inverse problems for semilinear elliptic PDE with a general nonlinearity $a(x,u)$
本文研究通过边界测量恢复椭圆方程中非线性项的问题，改进了基于一阶和高阶线性化的方法，证明一般非线性项在固定解附近唯一确定。
Inverse problems for semilinear Schrödinger equations at large frequency via polynomial resolvent estimates on manifolds
本文在大固定频率下研究一般紧致黎曼流形上半线性薛定谔方程的反边界问题，利用多项式预解估计、Gaussian beam拟模和驻相引理证明三次非线性项可由Dirichlet-to-Neumann映射唯一确定，无需乘积结构或测地射线变换单射性。
Bounded Time Inverse Scattering for Semilinear Dirac Equation
该研究针对带光滑非线性项的半线性Dirac方程，首次证明有界时间逆散射问题的唯一性，通过构造趋近初始时间线的碰撞序列模拟边界碰撞，无需假设非线性结构即可确定相关区域内的非线性项。
Stability estimates for inverse problems for semi-linear wave equations on Lorentzian manifolds
针对整体双曲Lorentz流形上的半线性波动方程，论文利用高阶线性化与Gaussian光束证明未知势可由DN映射Hölder稳定恢复，并放宽了边界凸性和类光测地线相交限制。
In this short note, we investigate simultaneous recovery inverse problems for semilinear elliptic equations with partial data. The main technique is based on higher order linearization and monotonicity approaches. With these methods at hand, we can determine the diffusion, cavity and coefficients simultaneously by knowing the corresponding localized Dirichlet-Neumann operators.
We consider the Dirichlet-to-Neumann map $Λ$ on a cylinder-like Lorentzian manifold related to the wave equation related to the metric $g$, a magnetic field $A$ and a potential $q$. We show that we can recover the jet of $g,A,q$ on the boundary from $Λ$ up to a gauge transformation in a stable way. We also show that $Λ$ recovers the following three invariants in a stable way: the lens relation of $g$, and the light ray transforms of $A$ and $q$. Moreover, $Λ$ is an FIO away from the diagonal with a canonical relation given by the lens relation. We present applications for recovery of $A$ and $…
We consider the sharp interface limit $ε\to 0$ of the semilinear wave equation $u_{tt} - Δu + \nabla W(u)/ ε^2 = 0$ in $\mathbf R^{1+n}$, where $u$ takes values in $\mathbf R^k$, $k = 1,2$, and $W$ is a double-well potential if $k = 1$ and vanishes on the unit circle and is positive elsewhere if $k = 2$. For fixed $ε> 0$ we find some special solutions, constructed around minimal surfaces in $\mathbf R^n$. In the general case, under some additional assumptions, we show that the solutions converge to a Radon measure supported on a time-like $k$-codimensional minimal submanifold of the Minkowski …
Rigidity of inverse problems for nonlinear elliptic equations on manifolds
针对带边界 Riemannian 流形上半线性椭圆方程的系数反问题，论文证明其一般不可行，并通过刚性表述证明：在适当条件下，DN 映射与线性方程一致即强制方程为线性。
This paper is devoted to the study of the existence of positive solutions for a problem related to a higher order fractional differential equation involving a nonlinear term depending on a fractional differential operator, $$(-Δ)^α u=λu+ (-Δ)^β|u|^{p-1}u \quad \mbox{in}\quad Ω;\qquad (-Δ)^{j}u=0\quad \mbox{on}\quad \partialΩ,\quad \mbox{for}\quad j\in\mathbb{Z},\: 0\leq j< [α]$$ where $Ω$ is a bounded domain in $\mathbb{R}^{N}$, $0<β<1$, $β<α<β+1$ and $λ>0$. In particular, we study the fractional elliptic problem, $$ (-Δ)^{α-β} u= λ(-Δ)^{-β}u+ |u|^{p-1}u \quad\mbox{in} \quad Ω;\qquad u=0 \quad…
We study global uniqueness in an inverse problem for the fractional semilinear Schrödinger equation $(-Δ)^{s}u+q(x,u)=0$ with $s\in (0,1)$. We show that an unknown function $q(x,u)$ can be uniquely determined by the Cauchy data set. In particular, this result holds for any space dimension greater than or equal to $2$. Moreover, we demonstrate the comparison principle and provide a $L^\infty$ estimate for this nonlocal equation under appropriate regularity assumptions.
Inverse scattering problems for non-linear wave equations on Lorentzian manifolds
该研究定义非线性散射泛函作为新型数据，利用波的非线性相互作用，在具有至少一个渐近闵可夫斯基型端的全局双曲开流形上求解了半线性波动方程逆散射问题，确定了流形的拓扑、微分结构、共形类型及度量和非线性系数的乘性变换形式，而线性波动方程对应逆问题仍未解决。
An inverse problem for a semilinear elliptic equation on conformally transversally anisotropic manifolds
本文研究了共形横向各向异性流形上的半线性椭圆方程反问题，通过分析高阶线性化及Gaussian准模解的相互作用，证明了利用Dirichlet-to-Neumann映射可唯一确定未知函数q，推广了已有结果。
An Inverse Source Problem for Semilinear Stochastic Hyperbolic Equations
针对半线性随机双曲方程的反源问题，提出结合Carleman估计与不动点迭代的全局收敛正则化方法，从部分横向Cauchy数据重建源函数，无需良好初值，并设计高效数值算法验证有效性。
Recovery of coefficients in semilinear transport equations
本文研究含时半线性输运方程的反问题，利用线性化方案与 Carleman 估计，证明线性及非线性时不变系数可由边界测量唯一稳定恢复，并覆盖欧几里得与一般几何情形。
We investigate the monotonicity method for fractional semilinear elliptic equations with power type nonlinearities. We prove that if-and-only-if monotonicity relations between coefficients and the derivative of the Dirichlet-to-Neumann map hold. Based on the strong monotonicity relations, we study a constructive global uniqueness for coefficients and inclusion detection for the fractional Calderón type inverse problem. Meanwhile, we can also derive the Lipschitz stability with finitely many measurements. The results hold for any $n\geq 1$.
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。
Stability for a formally determined Lorentzian inverse problem
本文研究洛伦兹流形上波动方程中时空依赖势的反演问题，在零锥叶状结构假设下证明Lipschitz稳定性与唯一性，方法为改进的Bukhgeim--Klibanov法结合畸变平面波。
In this paper we study the asymptotic behavior of solutions to an elliptic equation near the singularity of an inverse square potential with a coefficient related to the best constant for the Hardy inequality. Due to the presence of a borderline Hardy potential, a proper variational setting has to be introduced in order to provide a weak formulation of the equation. An Almgren-type monotonicity formula is used to determine the exact asymptotic behavior of solutions.
Determining Lorentzian manifold from non-linear wave observation at a single point
本文研究非线性双曲方程的反问题，证明通过单条类时曲线上的源到解映射可确定洛伦兹流形的共形结构，方法为非线性波相互作用的微局部分析。
Stackelberg-Nash strategy for the null controllability of semilinear degenerate equations in non-cylindrical domains
本研究针对非柱形域内一维半线性退化抛物方程的零能控问题，采用Stackelberg-Nash策略，结合退化非自治系统的Carleman不等式与Liusternik逆函数定理，证明了该系统的局部零能控性。
Starting with approximate solutions of the equation $-Δu=wu^3$ on the disk, with zero boundary conditions, we prove that there exist true solutions nearby. One of the challenges here lies in the fact that we need simultaneous and accurate control of both the (inverse) Dirichlet Laplacean and nonlinearities. We achieve this with the aid of a computer, using a Banach algebra of real analytic functions, based on Zernike polynomials. Besides proving existence, and symmetry properties, we also determine the Morse index of the solutions.
A time-dependent inverse source problem for a semilinear pseudo-parabolic equation with Neumann boundary condition
本文针对带Neumann边界条件的半线性伪抛物方程时变逆源问题，基于Rothe方法建立弱解存在唯一性，设计数值时间离散格式并证明收敛性，通过多项式正则化处理含噪数据，数值实验验证理论结果。
In this paper we show the small data solvability of suitable semilinear wave and Klein-Gordon equations on geometric classes of spaces, which include so-called asymptotically de Sitter and Kerr-de Sitter spaces, as well as asymptotically Minkowski spaces. These spaces allow general infinities, called conformal infinity in the asymptotically de Sitter setting; the Minkowski type setting is that of non-trapping Lorentzian scattering metrics introduced by Baskin, Vasy and Wunsch. Our results are obtained by showing the global Fredholm property, and indeed invertibility, of the underlying linear o…
We consider the inverse problem of determining a general semilinear term appearing in nonlinear parabolic equations. For this purpose, we derive a new criterion that allows to prove global recovery of some general class of semilinear terms from lateral boundary measurements of solutions of the equation with initial condition fixed at zero. More precisely, we prove, for what seems to be the first time, the unique and stable recovery of general semilinear terms depending on time and space variables independently of the solution of the nonlinear equation from the knowledge of the parabolic Dirich…
Rothe's method in direct and time-dependent inverse source problems for a semilinear pseudo-parabolic equation
本文利用罗特方法研究半线性伪抛物方程的反源问题，证明了解的存在性和唯一性，并提出数值方案验证其准确性与稳定性。
A reconstruction algorithm based on topological gradient for an inverse problem related to a semilinear elliptic boundary value problem
本文提出一种基于拓扑梯度的重构算法，用于解决与半线性椭圆偏微分方程相关的反问题，通过边界观测检测域内小不均匀性。
In this paper, we investigate the global behaviors of solutions to defocusing semilinear wave equations in $\mathbb{R}^{1+d}$ with $d\geq 3$. We prove that in the energy space the solution verifies the integrated local energy decay estimates for the full range of energy subcritical and critical power. For the case when $p>1+\frac{2}{d-1}$, we derive a uniform weighted energy bound for the solution as well as inverse polynomial decay of the energy flux through hypersurfaces away from the light cone. As a consequence, the solution scatters in the energy space and in the critical Sobolev space fo…
We consider the inverse problem of determining a general nonlinear term appearing in a semilinear hyperbolic equation on a Riemannian manifold with boundary $(M,g)$ of dimension $n=2,3$. We prove results of unique recovery of the nonlinear term $F(t,x,u)$, appearing in the equation $\partial_t^2u-Δ_gu+F(t,x,u)=0$ on $(0,T)\times M$ with $T>0$, from some partial knowledge of the solutions $u$ on the boundary of the time-space cylindrical manifold $(0,T)\times M$ or on the lateral boundary $(0,T)\times\partial M$. We determine the expression $F(t,x,u)$ both on the boundary $x\in\partial M$ and i…
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