Recovering the polytropic exponent in the porous medium equation: asymptotic approach

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发表日期：

2025年4月8日

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Hagop Karakazian,
Toni Sayah,
Faouzi Triki

摘要

In this paper we consider the time dependent Porous Medium Equation, $u_t = \Delta u^\gamma $ $u_{t} = Δ u^{γ}$ with real polytropic exponent $\gamma >1$ $γ > 1$ , subject to a homogeneous Dirichlet boundary condition. We are interested in recovering $\gamma $ $γ$ 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 $ $γ$ . An upper bound for the error between the exact and recovered $\gamma $ $γ$ is then showed. Finally, numerical investigations are carried out in two and three dimensions.