我们考虑时空$(M, g)$中的反问题，这是一个$4$维洛伦兹流形。对于半线性波动方程$\square_g u + H(x, u) = f$，其中$\square_g$表示通常的Laplace-Beltrami算子，我们证明了源-解映射$L: f \rightarrow u|_V$，其中$V$是类时测地线$μ$的邻域，决定了波可以从$μ$传播并返回的最大集合中时空度量的拓扑、可微结构和共形类。此外，在给定的时空$(M, g)$上，源到解映射确定$u$中的$H$的泰勒展开的一些系数。
英文摘要：
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 some coefficients of the Taylor expansion of $H$ in $u$.
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.
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 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.
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
University of Alaska at Fairbanks（阿拉斯加大学费尔班克斯分校）；Moscow Center of Fundamental and Applied Mathematics of Lomonosov Moscow State University（莫斯科国立大学基础与应用数学莫斯科中心）；University of Helsinki（赫尔辛基大学）
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.
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…
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…
An Inverse Almost Periodic Problem for a Semilinear Strongly Damped Wave Equation
Institute of Mathematics, Humboldt University of Berlin（柏林洪堡大学数学研究所）；Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine（乌克兰国家科学院应用力学与数学问题研究所）；Department of Computational Mathematics and Programming, Institute of Applied Mathematics and Fundamental Sciences, Lviv Polytechnic National University（利沃夫国立理工大学应用数学与基础科学研究所计算数学与编程系）；Institute of Mathematics, National Academy of Sciences of Ukraine（乌克兰国家科学院数学研究所）
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…
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…
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.
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 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 $n\geq 1$.
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.
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…
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…
