Geodesic
Recovery of the coefficient of $u_t$ in the heat equation from a condition of nonlocal observation in time
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 55 (2015) no. 1, pp. 89-104

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The inverse problem of finding the coefficient $\rho(x)=\rho_0+r(x)$ multiplying $u_t$ in the heat equation is studied. The unknown function $r(x)\geqslant0$ is sought in the class of bounded functions, and $\rho_0$ is a given positive constant. In addition to the initial and boundary conditions (data of the direct problem), a nonlocal observation condition is specified in the form $\int\limits_0^T u(x,t)d\mu(t)=\chi(x)$ with a given measure $d\mu(t)$ and a function $\chi(x)$. The case of integral observation (i.e., $d\mu(t)=\omega(t)dt$) is considered separately. Sufficient conditions for the existence and uniqueness of a solution to the inverse problem are obtained in the form of easy-to-check inequalities. Examples of inverse problems are given for which the assumptions of the theorems proved in this work are satisfied. 
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     author = {A. B. Kostin},
     title = {Recovery of the coefficient of $u_t$ in the heat equation from a condition of~nonlocal observation in time},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {89--104},
     publisher = {mathdoc},
     volume = {55},
     number = {1},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2015_55_1_a7/}
}
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A. B. Kostin. Recovery of the coefficient of $u_t$ in the heat equation from a condition of nonlocal observation in time. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 55 (2015) no. 1, pp. 89-104. http://geodesic.mathdoc.fr/item/ZVMMF_2015_55_1_a7/