On some classes of inverse problems for parabolic equations

Abstract

We study solvability of inverse problems of finding the right-hand side together with a solution itself for vector-valued parabolic equations. The usual boundary conditions are supplemented with the overdetermination conditions which are the values of a solution on some system of surfaces.

Hyperbolic Equations with Unknown Coefficients

“…In the first of them, it is assumed that the unknown coefficient does not depend on the time (distinguished) variable, i.e., in fact it depends on the space variables (see, for example, [1][2][3][4][5][6][7][8][9] and many other works); we refer to such inverse problems as space-type inverse problems. In the second case, it is, on the contrary, assumed that the unknown coefficient depends only on the time (distinguished) variable (see [1,[10][11][12][13][14][15][16][17]) and we will refer to such inverse problems as time-type inverse problems.…”

Inverse Problem for Source Function in Parabolic Equation at Neumann Boundary Conditions

“…It is necessary to mention that there are many known results concerning to the inverse problems close to the posed problem. Among of them it can be distinguish the coefficient inverse problems (see, e. g. [1][2][3]), problems with unknown source function [4][5][6] and problems, where unknown function in the boundary condition occurs [7]. The authors, dealing with finding the source function, usually assume, that this function is included into the equation by multiplicative way (see, for example, [4]).…”

“…The overdetermination conditions can be nonlocal integral ones [1,8,9]. One-point and twopoint overdetermination conditions are considered in [5,10]. As a rule, in the cited papers and books the existence and uniqueness of solution are proved, some asymptotic methods of solution construction are described.…”

On some classes of coefficient inverse problems for parabolic systems of equations with the ovedetermination date of evolutionary type

“…В этом случае неизвестные функции зависят только от , и поверхности -точки. Можно отметить работы [3], [4], где были рассмотрены задачи вида (1), (2) в общей поста-новке.…”