Recovery of the coefficient of u t in the heat equation from a condition of nonlocal observation in time

Abstract

The inverse problem of finding the coefficient ρ(x) = ρ 0 + r(x) multiplying u t in the heat equation is studied. The unknown function r(x) 0 is sought in the class of bounded functions, and ρ 0 is a given positive constant. In addition to the initial and boundary conditions (data of the direct problem), a nonlocal observation condition is specified in the form u(x, t)dμ(t) = χ(x) with a given measure dμ(t) and a function χ(x). The case of integral observation (i.e., dμ(t) = ω(t)dt) is considered separately. Sufficient conditions for the existence and uniqueness of a solution to the inverse problem are obtained in the form of easy to check inequalities. Examples of inverse problems are given for which the assumptions of the theorems proved in this work are satisfied.Keywords: coefficient inverse problems, inverse problem for the heat equation, nonlocal observation (or overdetermination) condition, sufficient conditions for the existence and uniqueness of a solution. ≥ 0 T ∫ COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS Vol. 55 No. 1 2015 KOSTINwhere μ(t) and χ(x) are given functions. Observation condition (1.2) is nonlocal in time t, and it was pro posed for a linear inverse problem in [19]. More specifically, it includes the final observation and the integral observation .If, for example, ω(t) = 1/T, then we have the condition of the integral mean of the function u(x, t):.Another typical example of a condition of form (1.2) is , where α k > 0 and t k are given numbers. This condition can be interpreted as setting a weighted mean value of u(x, t).The problem of determining the coefficient of u t in a parabolic equation from a final observation con dition was studied in [3,8]. The solution was considered in Hölder classes, which ruled out the possibility of discontinuous coefficients. In [6, 7] a similar coefficient inverse problem with final and integral obser vations was investigated in Sobolev classes. In [11] the coefficient of Δu was recovered in the case of inte gral observation. Note that the problem we consider below is not reduced to this case. In this paper, the results of [6,7] are extended to the more general overdetermination condition (1.2) and an iterative method for finding the coefficient r(x) in the class of bounded functions is proposed and justified. The case of integral observation for a smooth weight function ω(t), which has not been addressed earlier for such a problem, is considered separately. As a result, a unique solvability theorem for the inverse problem is proved under weaker constraints on the given functions. Note also that the inverse problem of recovering r(x) with a general observation of form (1.2), where μ(t) has a bounded variation on [0, T], has not been studied previously. However, even in the case of integral and final observations, Theorem 1 and 2 provide new (in comparison with [6-8]) unique solvability conditions for the inverse problem. Specifically, in contrast to [6][7][8], the initial data are no longer required to be zero, and an explicit estimate for the desir...

Determination of the coefficient function in a Whitham type nonlinear differential equation with impulse effects

“…Further, many papers appeared devoted to the study of questions of the unique solvability of the Cauchy problem for different types of partial differential equations of the first order (see, for example, [24][25][26][27][28][29][30][31][32][33]). The issues of determining the coefficient in various inverse problems have been considered by many authors, in particular, in [34][35][36][37][38][39].…”