Geodesic
On certain inverse problems for parabolic equations with final and integral observation
Sbornik. Mathematics, Tome 75 (1993) no. 2, pp. 473-490

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The inverse problems of finding the free term and the coefficient of $u(x,\,t)$ in a parabolic equation is considered. The Fredholm property of the linear inverse problem of finding the right-hand side of a special form is proved, along with global existence, uniqueness, and stability theorems for its solutions. A uniqueness theorem is established for the nonlinear inverse problem of determining the coefficient under restrictions in the form of inequalities containing no smallness conditions. The proof is carried out by a method of a priori estimates, with the use of the maximum principle for parabolic and elliptic equations. A connection between the uniqueness of the solution of the inverse problem and the completeness of a certain system of functions is established. 
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     author = {A. I. Prilepko and A. B. Kostin},
     title = {On certain inverse problems for parabolic equations with final and integral observation},
     journal = {Sbornik. Mathematics},
     pages = {473--490},
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     volume = {75},
     number = {2},
     year = {1993},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1993_75_2_a8/}
}
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A. I. Prilepko; A. B. Kostin. On certain inverse problems for parabolic equations with final and integral observation. Sbornik. Mathematics, Tome 75 (1993) no. 2, pp. 473-490. http://geodesic.mathdoc.fr/item/SM_1993_75_2_a8/