Inverse source problem for parabolic equation with the condition of integral observation in time

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 sufficient 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 fulfilled.

Inverse problem for semilinear Eidelman type equation

“…, and they satisfy equality (5) 2, (10). Then u * is a solution to the problem (1)-(3) with q * instead of q in (1).…”

“…On the other hand q * (t) and u * (z, t) satisfy (10), and therefore it is easy to get the following equality…”

“…Note, that the problems of determination of a parameter in the right-hand side function of the parabolic equations were studied in [7]- [12], of the semilinear ultraparabolic equations in [13,14,15]. The authors used the methods of the integral equations, regularization and the Shauder principle [7,8,10], the methods of finite difference approximations, numerical and iterative methods [11,12], the method of successive approximations [13]- [15].…”

Splitting method for an inverse source problem in parabolic differential equations: Error analysis and applications

“…Since the unbounded operator A generates an analytic semigroup e tA , then as a consequence of the semigroup theory, there exist M 0 and M 1 such that (15) and (16) are satisfied and by choosing a suitable equivalent norm ‖.‖, the bounds in assumptions (23) are concluded. Then, by using the general properties (smoothness, Lipschitz continuity) of nonlinear function F(t,Ũ(t)), the bounds (24) and (25) are obtained.…”

“…In these problems, the source parameter can be determined with the additional overspecified data. We can also find other examples in which the integral condition (2) arises, for an instance, heat transmission in a thin rod in particle diffusion in turbulent plasma (see, e.g., [9][10][11][12][13][14][15]).…”