J. Inverse Ill-Posed Probl. 2017; aop Research Article Aleksey I. Prilepko, Vitaly L. Kamynin and Andrew B. Kostin* Inverse source problem for parabolic equation with the condition of integral observation in time DOI: 10.1515/jiip-2017-0049 Received May 5, 2017; revised August 23, 2017; accepted September 1, 2017 Abstract: We consider the inverse problem of source determination in nonuniformly parabolic equation under the additional condition of integral observation. We investigate the questions of existence and uniqueness of solution. Two types of sufcient conditions for the unique solvability of the inverse problem are obtained. Examples of inverse problems are given for which the conditions of the proved theorems are fulőlled. Keywords: Inverse problems, degenerate parabolic equations, integral observation MSC 2010: 35K20, 35R30 || Dedicated to Anatoly Bakushinsky on the occasion of his 80th birthday 1 Introduction The purpose of our study will be the questions of existence and uniqueness of solution {u(t , x); p(x)} of inverse problems for nonuniformly parabolic equations with additional condition of integral observation T ∫ 0 u(t , x)χ(t) dt = φ(x), x ∈ Ω. (1.1) For the őrst time, the inverse problems for nonstationary diferential equations with condition (1.1) were considered in the paper [25] for uniformly parabolic equations u t − n ∑ i , j=1 ∂ ∂x i (a ij (x) ∂u ∂x j )− n ∑ i=1 b i (x) ∂u ∂x i − c(x)u = g(t , x)p(x)+ r(t , x), and in the papers [26, 30] for evolution equations in a Banach space: u  (t)= Au(t)+ Φ(t)p. Further, the inverse problems of determination of the unknown right-hand side and the unknown coef- őcients in parabolic equations with the additional condition (1.1) were considered by many authors (see, e.g., [7, 8, 12, 15, 17, 18, 27, 28, 31, 32]), but mainly for uniformly parabolic equations. On the other hand, the inverse problems for degenerate parabolic equations are important in applications; in particular, they Aleksey I. Prilepko: Department of Mathematical Analysis, Lomonosov Moscow State University, Leninskie Gory 1, 119991 Moscow, Russia, e-mail: prilepko.ai@yandex.ru Vitaly L. Kamynin: Department of Mathematics, National Research Nuclear University MEPhI, Kashirskoe Shosse 31, 115409 Moscow, Russia, e-mail: vlkamynin2008@yandex.ru *Corresponding author: Andrew B. Kostin: Department of Mathematics, National Research Nuclear University MEPhI, Kashirskoe Shosse 31, 115409 Moscow, Russia, e-mail: abkostin@yandex.ru. http://orcid.org/0000-0002-0929-1778 Brought to you by | University of Gothenburg Authenticated Download Date | 10/9/17 5:56 AM