PUB - Publikationen an der Universität Bielefeld

In this paper we establish uniqueness in the inverse boundary value problem for the porous medium equation & varepsilon;partial derivative(t)u-del & sdot;(gamma del u(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 & varepsilon; 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 ->infinity, of the solutions to the transformed equation, with special boundary data, allows us to obtain sufficient information to deduce the uniqueness result.