Recovering the Polytropic Exponent in the Porous Medium Equation: Asymptotic Approach

In this paper we consider the time dependent Porous Medium Equation, $u_t=\mathrm{\Delta }{u}^{\gamma }$ with real polytropic exponent $\gamma >1$, subject to a homogeneous Dirichlet boundary condition. We are interested in recovering $\gamma$ from the knowledge of the solution $u$ at a given large time $T$. Based on an asymptotic inequality satisfied by the solution $u(T)$, we propose a numerical algorithm allowing us to recover $\gamma$. An upper bound for the error between the exact and recovered $\gamma$ is then showed. Finally, numerical investigations are carried out in two dimensions.