FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2006, VOLUME 12, NUMBER 4, PAGES 187-202

Certain inverse problems for parabolic equations

S. G. Pyatkov

Abstract

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In the paper, we study the inverse problem of finding the solution u and the coefficient q from the following data:

Mu = ut - L(x,t,Dx)u + g(x,t,u,Ñu) + q(x,t)u(x,t) = f(x,t),
(x,t) Î Q = G ´ (0,T),
u|S = f(x,t),   (u)/(n)|S = y(x,t),   u|t=0 = u0(x),   S = G ´ (0,T),

where G Ì Rn is a bounded domain with boundary G and L is a second-order elliptic operator. We prove that the problem is solvable locally in time or in the case where the norms of its data are sufficiently small.

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