On uniqueness and reconstruction of a nonlinear diffusion term in a parabolic equation

The problem of recovering coefficients in a diffusion equation is one of the basic inverse problems. Perhaps the most important term is the one that couples the length and time scales and is often referred to as {\it the\/} diffusion coefficient $a$ in $u_t - \nabla(a\nabla u) = f$. In this paper we seek the unknown $a$ assuming that $a=a(u)$ depends only on the value of the solution at a given point. Such diffusion models are the basic of a wide range of physical phenomena such as nonlinear heat conduction, chemical mixing and population dynamics. We shall look at two types of overposed data in order to effect recovery of $a(u)$: the value of a time trace $u(x_0,t)$ for some fixed point $x_0$ on the boundary of the region $Ω$; or the value of $u$ on an interior curve $Σ$ lying within $Ω$. As examples, these might represent a temperature measurement on the boundary or a census of the population in some subset of $Ω$ taken at a fixed time $T>0$. In the latter case we shall show a uniqueness result that leads to a constructive method for recovery of $a$. Indeed, for both types of measured data we shall show reconstructions based on the iterative algorithms developed in the paper.

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