{"title":"通过事故后观测确定扩散方程和波动方程中的源项：唯一性和稳定性","authors":"Jin Cheng, Shuai Lu, Masahiro Yamamoto","doi":"10.4208/csiam-am.so-2022-0028","DOIUrl":null,"url":null,"abstract":"We consider a diffusion and a wave equations: $$ \\partial_t^ku(x,t) = \\Delta u(x,t) + \\mu(t)f(x), \\quad x\\in \\Omega, \\, t>0, \\quad k=1,2 $$ with the zero initial and boundary conditions, where $\\Omega \\subset \\mathbb{R}^d$ is a bounded domain. We establish uniqueness and/or stability results for inverse problems of 1. determining $\\mu(t)$, $0<t<T$ with given $f(x)$; 2. determining $f(x)$, $x\\in \\Omega$ with given $\\mu(t)$ \\end{itemize} by data of $u$: $u(x_0,\\cdot)$ with fixed point $x_0\\in \\Omega$ or Neumann data on subboundary over time interval. In our inverse problems, data are taken over time interval $T_1<t<T_1$, by assuming that $T<T_1<T_2$ and $\\mu(t)=0$ for $t\\ge T$, which means that the source stops to be active after the time $T$ and the observations are started only after $T$. This assumption is practical by such a posteriori data after incidents, although inverse problems had been well studied in the case of $T=0$. We establish the non-uniqueness, the uniqueness and conditional stability for a diffusion and a wave equations. The proofs are based on eigenfunction expansions of the solutions $u(x,t)$, and we rely on various knowledge of the generalized Weierstrass theorem on polynomial approximation, almost periodic functions, Carleman estimate, non-harmonic Fourier series.","PeriodicalId":29749,"journal":{"name":"CSIAM Transactions on Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":1.2000,"publicationDate":"2021-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"Determination of Source Terms in Diffusion and Wave Equations by Observations After Incidents: Uniqueness and Stability\",\"authors\":\"Jin Cheng, Shuai Lu, Masahiro Yamamoto\",\"doi\":\"10.4208/csiam-am.so-2022-0028\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We consider a diffusion and a wave equations: $$ \\\\partial_t^ku(x,t) = \\\\Delta u(x,t) + \\\\mu(t)f(x), \\\\quad x\\\\in \\\\Omega, \\\\, t>0, \\\\quad k=1,2 $$ with the zero initial and boundary conditions, where $\\\\Omega \\\\subset \\\\mathbb{R}^d$ is a bounded domain. We establish uniqueness and/or stability results for inverse problems of 1. determining $\\\\mu(t)$, $0<t<T$ with given $f(x)$; 2. determining $f(x)$, $x\\\\in \\\\Omega$ with given $\\\\mu(t)$ \\\\end{itemize} by data of $u$: $u(x_0,\\\\cdot)$ with fixed point $x_0\\\\in \\\\Omega$ or Neumann data on subboundary over time interval. In our inverse problems, data are taken over time interval $T_1<t<T_1$, by assuming that $T<T_1<T_2$ and $\\\\mu(t)=0$ for $t\\\\ge T$, which means that the source stops to be active after the time $T$ and the observations are started only after $T$. This assumption is practical by such a posteriori data after incidents, although inverse problems had been well studied in the case of $T=0$. We establish the non-uniqueness, the uniqueness and conditional stability for a diffusion and a wave equations. The proofs are based on eigenfunction expansions of the solutions $u(x,t)$, and we rely on various knowledge of the generalized Weierstrass theorem on polynomial approximation, almost periodic functions, Carleman estimate, non-harmonic Fourier series.\",\"PeriodicalId\":29749,\"journal\":{\"name\":\"CSIAM Transactions on Applied Mathematics\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2021-07-17\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"CSIAM Transactions on Applied Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.4208/csiam-am.so-2022-0028\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"CSIAM Transactions on Applied Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4208/csiam-am.so-2022-0028","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Determination of Source Terms in Diffusion and Wave Equations by Observations After Incidents: Uniqueness and Stability
We consider a diffusion and a wave equations: $$ \partial_t^ku(x,t) = \Delta u(x,t) + \mu(t)f(x), \quad x\in \Omega, \, t>0, \quad k=1,2 $$ with the zero initial and boundary conditions, where $\Omega \subset \mathbb{R}^d$ is a bounded domain. We establish uniqueness and/or stability results for inverse problems of 1. determining $\mu(t)$, $0
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