On a stability estimate for a solution to an~inverse problem for a hyperbolic equation
type: Item
instance of: scholarly article
title: On a stability estimate for a solution to an~inverse problem for a hyperbolic equation
zbMATH Open document ID: 0938.35191
DOI: 10.1007/BF02677522
published in: Siberian Mathematical Journal
publication date: 2000-04-13T00:00:00Z
review text: Suppose that a function \(u(x,t)\) is a solution to the Cauchy problem \[ L_q u\equiv u_{tt}-\Delta u -q(x) u =8\pi\delta (x,t), \quad u|_{t<0} \equiv 0, \quad (x, t)\in {\mathbb R}^4, \] where \(\delta (x,t)\) is Dirac's delta-function supported at the point \((0,0)\). Let \(D\subset B(x^0,R)\) \(=\{x\in {\mathbb R}^3: |x^0-x|<R\}\), \(S_T=\{(x,t):\;x\in\partial D,|x|<t<T+|x|\}\), and \(|x^0|> R\). Suppose that the traces of the function \(u(x,t)\) and its first-order derivatives are known on the surface \(S_T\): \[ D^\alpha u\left|_{S_T}=f^\alpha (x,t),\right. \quad \alpha=(\alpha_1,\alpha_2, \alpha_3, \alpha_4), \quad |\alpha|\leq 1. \] The inverse problem is to reconstruct the coefficient \(q(x)\) for \(x\in D\). The author establishes stability estimates for \(q(x)\) and a uniqueness theorem for a solution to the inverse problem. In particular, the conditions guaranteeing stability are \(|x^0|>\sqrt{2}R\), \(4R/T\in(0,1)\), or a smallness condition on the number \(R\).
Mathematics Subject Classification ID: 35R30
Mathematics Subject Classification ID: 35L05
zbMATH DE Number: 1431678
zbMATH Keywords: hyperbolic equation
zbMATH Keywords: inverse problem
zbMATH Keywords: stability estimate
zbMATH Keywords: uniqueness theorem
author: Vladimir G. Romanov
MaRDI profile type: Publication
cites work: Q5676616
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full work available at URL: https://doi.org/10.1007/bf02677522
OpenAlex ID: W2015082080
description: scientific article; zbMATH DE number 1431678