This paper addresses the time-dependent Porous Medium Equation, $u_{t} - \alpha \Delta u^{\gamma }= 0$ with polytropic exponent $\gamma >1$ and diffusion coefficient $\alpha >0$ . Given the value of $\gamma $ and the solution $u$ at a large time $T$ , our goal is to determine the parameter $\alpha $ without the knowledge of the initial data $u(0)$ . Leveraging an asymptotic inequality satisfied by $u(T)$ , we propose a numerical algorithm to recover $\alpha $ through a minimization problem. Furthermore, we establish an upper bound on the error between the exact and recovered values of $\alpha $ and perform numerical simulations in two and three dimensional cases.
