On solvability of an inverse problem with an unknown coefficient and right-hand side for a parabolic equation

Abstract

We study the inverse problem for a parabolic equation in which the unknowns are the solution, coefficient of the function, and right-hand side (exterior load). We prove existence and uniqueness theorems.Let D be a bounded domain in R n with boundary Γ and let Q be the cylinder D × (0, T ), 0 < T < +∞. Consider the following equation in Q:where λ is a given positive constant, f (x, t) and a(x, t) are given functions subject to the conditions to be specified below, and q(x) and q 0 (x) are functions to be determined together with a solution u(x, t).The problems of this kind are called inverse, and there are many articles devoted to their study. Among the latter the most closed to us are [1-17]. The problem under consideration relates to the class of problems in which the unknowns are not only a solution but also the coefficient of the equation and right-hand side. The fact that the coefficient is unknown makes this problem nonlinear.We turn to the exact statement of the problem. Considering inverse problems, we must in general provide extra information or take extra measurements alongside the initial information corresponding to the information about boundary values of some direct boundary value problem (i.e., a problem with available coefficients and right-hand side). In our case we suggest using as complementary the information on the state and velocity of the medium at a fixed point of time.

Simultaneous identification and reconstruction of the space-dependent reaction coefficient and source term

“…We utilise the methodology applied in [26], see also [24,25], and consider the following two auxiliary parabolic problems:…”

Solvability of the Inverse Problem of Finding Thermal Conductivity

“…The conditions (12) and (13) of the theorem look rather hard to check; moreover, it is not obvious that the set of the initial data is nonempty, i.e., the set of the functions f (x, t), u 0 (x), u 1 (x), µ 0 (t), and µ 1 (t) and the numbers T which satisfy (8)- (15). We give an example of a situation in which checking all necessary conditions is simple and we can easily verify that the set of input data of the inverse problem (1)-(4) which satisfy all necessary conditions is nonempty.…”

“…(3) unknown coefficient ρ(x) [10,11]; (4) unknown coefficient q(x) and right-hand side [14,15]; (5) unknown coefficients ρ(x) and q(x) [16]. The problems of finding the unknown coefficient p(x) (together with the solution) in the frame of the inverse problems under consideration with final or integral overdetermination were studied earlier.…”

Inverse Problem of Finding the Coefficient of the Lowest Term in Two-Dimensional Heat Equation with Ionkin-Type Boundary Condition

“…Among these inverse problems, much attention is given to the determination of the lowest order coefficient in heat equation, in particular, when this coefficient depends solely on time. Various methods for finding the lowest order coefficient in a more general multidimensional parabolic equation have been addressed in numerous works, see [4,5,6,7,8] for time dependent coefficient, [9,10,11,12,13,14,15] for space dependent coefficient, [16,17,18] for both time and space dependent coefficient. The boundary conditions are most frequently classical (Dirichlet, Neumann and Robin) and additional condition is most frequently specified as the solution at an interior point or an integral mean over the entire domain.…”