We are concerned with a degenerate in time first-order identification problem related to a closed operator in a Banach space. The degeneracy with respect to time — due to scalar time coefficients — is assumed to be integrable. For both direct and inverse problem we exhibit explicit representations of the solutions in terms of the linear operator A and function ƒ (cf. equation (1.1)), when the latter possess specific properties.
Some applications to partial differential equations are given.
We recover a non-negative time-dependent function, vanishing only at t = 0, in a linear multi-dimensional parabolic equation with a non-integrable degeneration concentrated at t = 0. First we prove a global in time existence and uniqueness result for the direct problem in a general Hilbert space, via Fourier representation of the solution to the direct problem. Then we show a similar result, but local in time only, for the identification problem. Finally, we apply such a result to our specific linear parabolic equation related to a smooth bounded domain in ℝd, d = 1, 2, 3.
- We are concerned with the problem of recovering the radial kernel k, depending also on time, in the parabolic integro-differential equation
, A being a uniformly elliptic second-order linear operator in divergence form. We single out a special class of operators A and two pieces of suitable additional information for which the problem of identifying k can be uniquely solved locally in time when the domain under consideration is a spherical corona or an annulus.
We are concerned with the problem of recovering the kernel k, depending on time and having a special spatial symmetry, in the parabolic integro-differential equation (1.1) and related to a domain Ω which is union of level sets of each function k(t, ·). We single out a special class of differential operators A and two pieces of suitable additional information for which the problem of identifying k can be uniquely solved locally in time.
As is well known, the propagation of electromagnetic waves in dispersive media is governed by integro-differential equations. We assume here that the medium is a rigid body with a cylindric symmetry. In this case all the physical characteristics, such as the dielectric coefficient, the magnetic permeability and the conductivity coefficient as well as the kernels accounting for memory effects, may be assumed to depend only on the distance from the axis of the cylinder.
Our aim is to solve the inverse problem, consisting in determining, in addition to the electromagnetic field, also the relaxation kernels, by the means of additional measurements.
Existence, uniqueness and continuous dependence results are proved in the context of suitable functional spaces.
In this paper we determine, under a suitable additional information and in a framework of Gevrey-type functions with respect to the variable x1, the spatial part p(x1, x3) of the factorised kernel σ1 (x1, x3, t) = p(x1, x3)k(t) in the integrodifferential Maxwell system related to a spatial domain of the form Ω × ×
+, where Ω is an interval in . In our context determining p means to show locally in space existence, uniqueness and continuous dependence of p on the data.