Volume 4, Issue 2
Determination of Source Terms in Diffusion and Wave Equations by Observations After Incidents: Uniqueness and Stability

Jin Cheng, Shuai Lu & Masahiro Yamamoto

CSIAM Trans. Appl. Math., 4 (2023), pp. 381-418.

Published online: 2023-02

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  • Abstract

We consider a diffusion and a wave equations

$$∂^k_tu(x,t) =∆u(x,t)+\mu(t)f(x), x∈Ω, t>0, k=1,2$$

with the zero initial and boundary conditions, where $Ω ⊂\mathbb{R}^d$ is a bounded domain. We establish uniqueness and/or stability results for inverse problems of determining $\mu(t), 0<t< T$ with given $f(x),$ determining $f(x), x ∈ Ω$ with given $\mu(t),$ by data of $u: u(x_0,·)$ with fixed point $x_0 ∈ Ω$ or Neumann data on subboundary over time interval. In our inverse problems, data are taken over time interval $T_1<t<T_2,$ by assuming that $T < T_1 < T_2$ and $\mu(t) =0$ for $t≥ T,$ 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.

  • AMS Subject Headings

35R30, 35R25, 35K20, 35L20

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CSIAM-AM-4-381, author = {Cheng , JinLu , Shuai and Yamamoto , Masahiro}, title = {Determination of Source Terms in Diffusion and Wave Equations by Observations After Incidents: Uniqueness and Stability}, journal = {CSIAM Transactions on Applied Mathematics}, year = {2023}, volume = {4}, number = {2}, pages = {381--418}, abstract = {

We consider a diffusion and a wave equations

$$∂^k_tu(x,t) =∆u(x,t)+\mu(t)f(x), x∈Ω, t>0, k=1,2$$

with the zero initial and boundary conditions, where $Ω ⊂\mathbb{R}^d$ is a bounded domain. We establish uniqueness and/or stability results for inverse problems of determining $\mu(t), 0<t< T$ with given $f(x),$ determining $f(x), x ∈ Ω$ with given $\mu(t),$ by data of $u: u(x_0,·)$ with fixed point $x_0 ∈ Ω$ or Neumann data on subboundary over time interval. In our inverse problems, data are taken over time interval $T_1<t<T_2,$ by assuming that $T < T_1 < T_2$ and $\mu(t) =0$ for $t≥ T,$ 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.

}, issn = {2708-0579}, doi = {https://doi.org/10.4208/csiam-am.SO-2022-0028}, url = {http://global-sci.org/intro/article_detail/csiam-am/21419.html} }
TY - JOUR T1 - Determination of Source Terms in Diffusion and Wave Equations by Observations After Incidents: Uniqueness and Stability AU - Cheng , Jin AU - Lu , Shuai AU - Yamamoto , Masahiro JO - CSIAM Transactions on Applied Mathematics VL - 2 SP - 381 EP - 418 PY - 2023 DA - 2023/02 SN - 4 DO - http://doi.org/10.4208/csiam-am.SO-2022-0028 UR - https://global-sci.org/intro/article_detail/csiam-am/21419.html KW - Inverse source problem, data after incidents, diffusion equation, wave equation. AB -

We consider a diffusion and a wave equations

$$∂^k_tu(x,t) =∆u(x,t)+\mu(t)f(x), x∈Ω, t>0, k=1,2$$

with the zero initial and boundary conditions, where $Ω ⊂\mathbb{R}^d$ is a bounded domain. We establish uniqueness and/or stability results for inverse problems of determining $\mu(t), 0<t< T$ with given $f(x),$ determining $f(x), x ∈ Ω$ with given $\mu(t),$ by data of $u: u(x_0,·)$ with fixed point $x_0 ∈ Ω$ or Neumann data on subboundary over time interval. In our inverse problems, data are taken over time interval $T_1<t<T_2,$ by assuming that $T < T_1 < T_2$ and $\mu(t) =0$ for $t≥ T,$ 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.

Jin Cheng, Shuai Lu & Masahiro Yamamoto. (2023). Determination of Source Terms in Diffusion and Wave Equations by Observations After Incidents: Uniqueness and Stability. CSIAM Transactions on Applied Mathematics. 4 (2). 381-418. doi:10.4208/csiam-am.SO-2022-0028
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