Recovering the Diffusion Coefficient in the Porous Medium Equation from a Large-Time Measurement

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.