Geodesic
The inverse problem of recovering the source in a~parabolic equation under a~condition of nonlocal observation
Sbornik. Mathematics, Tome 204 (2013) no. 10, pp. 1391-1434

Voir la notice de l'article provenant de la source Math-Net.Ru

We study the inverse problem for a parabolic equation of recovering the source, that is, the right-hand side $F(x,t)=h(x,t)f(x)$, where the function $f(x)$ is unknown. To find $f(x)$, along with the initial and boundary conditions, we also introduce an additional condition of nonlocal observation of the form $\displaystyle\int_{0}^{T}u(x,t)\,d\mu(t)=\chi(x)$. We prove the Fredholm property for the problem stated in this way, and obtain sufficient conditions for the existence and uniqueness of a solution. These conditions are of the form of readily verifiable inequalities and put no restrictions on the value of $T>0$ or the diameter of the domain $\Omega$ under consideration. The proof uses a priori estimates and the qualitative properties of solutions of initial-boundary value problems for parabolic equations. Bibliography: 40 titles.
Keywords: inverse problems, nonlocal overdetermination.
Mots-clés : parabolic equations 
@article{SM_2013_204_10_a0,
     author = {A. B. Kostin},
     title = {The inverse problem of recovering the source in a~parabolic equation under a~condition of nonlocal observation},
     journal = {Sbornik. Mathematics},
     pages = {1391--1434},
     publisher = {mathdoc},
     volume = {204},
     number = {10},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2013_204_10_a0/}
}
TY  - JOUR
AU  - A. B. Kostin
TI  - The inverse problem of recovering the source in a~parabolic equation under a~condition of nonlocal observation
JO  - Sbornik. Mathematics
PY  - 2013
SP  - 1391
EP  - 1434
VL  - 204
IS  - 10
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SM_2013_204_10_a0/
LA  - en
ID  - SM_2013_204_10_a0
ER  - 
%0 Journal Article
%A A. B. Kostin
%T The inverse problem of recovering the source in a~parabolic equation under a~condition of nonlocal observation
%J Sbornik. Mathematics
%D 2013
%P 1391-1434
%V 204
%N 10
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SM_2013_204_10_a0/
%G en
%F SM_2013_204_10_a0
A. B. Kostin. The inverse problem of recovering the source in a~parabolic equation under a~condition of nonlocal observation. Sbornik. Mathematics, Tome 204 (2013) no. 10, pp. 1391-1434. http://geodesic.mathdoc.fr/item/SM_2013_204_10_a0/