AN INVERSE BOUNDARY VALUE PROBLEM FOR THE INHOMOGENEOUS POROUS MEDIUM EQUATION

In this paper we establish uniqueness in the inverse boundary value problem for the porous medium equation \epsilon\partialtu - \nabla \cdot (\gamma\nablau^m) = 0, which is a degenerate parabolic type quasilinear PDE. We assume that m > 1, which is sometimes referred to as the slow diffusion case. Under these assumptions we show that the corresponding Dirichlet-to-Neumann map determines the two coefficients \epsilon and \gamma. Our approach relies on using a Laplace transform to turn the original equation into a coupled family of nonlinear elliptic equations, indexed by the frequency parameter (1/h in our definition) of the transform. A careful analysis of the asymptotic expansion in powers of h, as h \rightarrow \infty, of the solutions to the transformed equation, with special boundary data, allows us to obtain sufficient information to deduce the uniqueness result.