2022
DOI: 10.1016/j.jde.2022.08.010
|View full text |Cite
|
Sign up to set email alerts
|

Uniqueness, reconstruction and stability for an inverse problem of a semi-linear wave equation

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...
3
2

Citation Types

0
7
0

Year Published

2023
2023
2025
2025

Publication Types

Select...
6
1

Relationship

0
7

Authors

Journals

citations
Cited by 16 publications
(7 citation statements)
references
References 42 publications
0
7
0
Order By: Relevance
“…First, introduce some notations and assumptions. Assume that m$m$ is a positive integer and define the energy space Em$E^m$ as in [36, section 1] and [2, Definition 3.5 on p. 596]: Em=k=0mCkfalse([0,T];Hmk(Ω)false),$$\begin{align} E^m = \bigcap _{ k=0 }^m C^k ([0,T];H^{m-k}(\Omega)), \end{align}$$which is equipped with the norm ·Em$\Vert \cdot \Vert _{E^m}$ as uEm=trueprefixsup0tTk=0mtkufalse(·,tfalse)true∥Hmkfalse(normalΩfalse),uEm.$$\begin{align*} \Vert u \Vert _{E^m}=\sup _{0\leqslant t\leqslant T} \sum _{k=0}^m \Big \Vert \partial _t ^k u(\cdot, t)\Big \Vert _{H^{m-k}(\Omega)}, \quad \forall \ u\in E^m. \end{align*}$$…”
Section: Statement Of Main Resultsmentioning
confidence: 99%
See 2 more Smart Citations
“…First, introduce some notations and assumptions. Assume that m$m$ is a positive integer and define the energy space Em$E^m$ as in [36, section 1] and [2, Definition 3.5 on p. 596]: Em=k=0mCkfalse([0,T];Hmk(Ω)false),$$\begin{align} E^m = \bigcap _{ k=0 }^m C^k ([0,T];H^{m-k}(\Omega)), \end{align}$$which is equipped with the norm ·Em$\Vert \cdot \Vert _{E^m}$ as uEm=trueprefixsup0tTk=0mtkufalse(·,tfalse)true∥Hmkfalse(normalΩfalse),uEm.$$\begin{align*} \Vert u \Vert _{E^m}=\sup _{0\leqslant t\leqslant T} \sum _{k=0}^m \Big \Vert \partial _t ^k u(\cdot, t)\Big \Vert _{H^{m-k}(\Omega)}, \quad \forall \ u\in E^m. \end{align*}$$…”
Section: Statement Of Main Resultsmentioning
confidence: 99%
“…Inspired by [36, Definition 1], we impose the following conditions on f$\tilde{f}$. Definition For any T>T$T>T^*$ (in ()), truef=truef(x,t,s)$\tilde{f}=\tilde{f}(x,t,s)$:Q×double-struckRdouble-struckR$: Q\times \mathbb {R}\rightarrow \mathbb {R}$ is called an admissible coefficient, if it satisfies the following: (1)Analyticity on R$\mathbb {R}$: {the mapstruef(·,·,s)is analytic onRwith values inEm+1,ffalse(x,t,0false)=0,inQ.$$\begin{align} {\begin{cases} \mbox{ the map $ s\mapsto \tilde{f}(\cdot,\cdot, s)$ is analytic on $\mathbb {R}$ with values in $E^{m+1}$}, \\ \ \tilde{f}(x,t,0)=0, \mbox{ in } Q.…”
Section: Statement Of Main Resultsmentioning
confidence: 99%
See 1 more Smart Citation
“…This is a central question in the study of inverse problems for wave equations, which is a subject that has attracted considerable attention in recent years (see e.g. [2,3,8,12,13,14,17,19,23,24] and references therein).…”
Section: Introductionmentioning
confidence: 99%
“…In the case of (genuinely) semilinear wave equations, it has been shown in various settings that, up to natural obstructions, one can indeed recover the metric from the Dirichlet-to-Neumann map [10,12,13,19]. However, the corresponding problems for linear equations are still largely open.…”
Section: Introductionmentioning
confidence: 99%