An inverse problem for the porous medium equation with partial data and a possibly singular absorption term

TL;DR

This paper establishes the uniqueness of recovering coefficients in a porous medium equation with absorption from partial boundary data, even with finite observation time, advancing inverse problem theory for degenerate parabolic PDEs.

Contribution

It proves the unique identifiability of three coefficients in a degenerate parabolic PDE with absorption, using partial boundary data and finite time observations, including the case with a singular absorption term.

Findings

Uniqueness of coefficient recovery with partial boundary data.

Finite time observation suffices for global uniqueness.

Inclusion of absorption term enhances inverse problem results.

Abstract

In this paper we prove uniqueness in the inverse boundary value problem for the three coefficient functions in the porous medium equation with an absorption term $\epsilon\partial_t u-\nabla\cdot(\gamma\nabla u^m)+\lambda u^q=0$, with $m>1$, $m^{-1}<q<\sqrt{m}$, with the space dimension 2 or higher. This is a degenerate parabolic type quasilinear PDE which has been used as a model for phenomena in fields such as gas flow (through a porous medium), plasma physics, and population dynamics. In the case when $\gamma=1$ a priori, we prove unique identifiability with data supported in an arbitrarily small part of the boundary. Even for the global problem we improve on previous work by obtaining uniqueness with a finite (rather than infinite) time of observation and also by introducing the additional absorption term $\lambda u^q$.