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

Published online: 2023-02

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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} }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.

*CSIAM Transactions on Applied Mathematics*.

*4*(2). 381-418. doi:10.4208/csiam-am.SO-2022-0028