ABSTRACT
In this paper we discuss some strategies we can use in the study of parabolic integrodifferential inverse problems. The choice of the strategy depends on what type of nonlinearities are involved. We consider the heat equation for materials with memory since it is one of the most important physical examples to which our methods apply. Other models, for instance in the theory of population dynamics, can also be considered within our framework. We recall, for the sake of completeness, the heat equation for materials with memory. Let Ω be an open and bounded set in R3 and T be a positive real number. The evolution equation for the temperature u is given, for (t, x) ∈ [0, T ]× Ω, by
Dtu(t, x) = k∆u(t, x) + ∫ t 0
h(t− s)∆u(s, x) ds+ F (u(t, x)), (1.1)
where k is the diffusivity coefficient, h accounts for the memory effects and F is the heat source. In the inverse problem we consider, besides the temperature u, also h as a further unknown, and to determine it we add an additional measurement on u represented in integral form by∫
φ(x)u(t, x) dx = G(t), ∀t ∈ [0, T ], (1.2)
where φ and G are given functions representing the type of device used to measure u (on a suitable part of the body Ω) and the result of the measurement, respectively. We associate with (1.1)–(1.2) the initial-boundary conditions, for example of Neumann type:{
u(0, x) = u0(x), x ∈ Ω, Dνu(t, x) = 0, (t, x) ∈ [0, T ]× ∂Ω,
(1.3)
ν denoting the outward normal unit vector. So one of the problems we are going to investigate is the following.
