AN INVERSE PROBLEM FOR THE POROUS MEDIUM EQUATION WITH PARTIAL DATA AND A POSSIBLY SINGULAR ABSORPTION TERM

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 є∂tu - ∇ ∙ (γ∇u^m) + λu^q = 0 with m > 1, m ^-1 < q < √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 γ = 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 λu^q.