In this paper we consider the time dependent Porous Medium Equation, $u_t = \Delta u^\gamma$ 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.

翻译：本文研究了含实数多方指数$\gamma>1$的时变多孔介质方程$u_t = \Delta u^\gamma$，并考虑齐次Dirichlet边界条件。我们感兴趣的是通过给定大时间$T$时解$u$的信息来恢复$\gamma$。基于解$u(T)$满足的渐近不等式，我们提出了一种能够恢复$\gamma$的数值算法。随后，给出了精确$\gamma$与恢复$\gamma$之间误差的上界。最后，在二维情形下进行了数值研究。