Recovering the elapsed time in the porous medium equation from a single large-time measurement

Consider the Porous Medium Equation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msub> <m:mo>∂</m:mo> <m:mi>t</m:mi> </m:msub> <m:mo>⁡</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mi mathvariant="normal">Δ</m:mi> <m:mo>⁢</m:mo> <m:msup> <m:mi>u</m:mi> <m:mi>γ</m:mi> </m:msup> </m:mrow> </m:mrow> </m:math> {∂ₜu=Δ uγ} subject to homogeneous Dirichlet boundary condition, where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>γ</m:mi> <m:mo>></m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {γ>1} is known as the polytropic exponent. Given the values of γ and the solution <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>u</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>T</m:mi> <m:mi>e</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {u(Tₑ)} at a sufficiently large but unknown time <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>T</m:mi> <m:mi>e</m:mi> </m:msub> </m:math> {Tₑ} , in this paper we provide an explicit formula for recovering the elapsed time <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>T</m:mi> <m:mi>e</m:mi> </m:msub> </m:math> {Tₑ} , without any knowledge of the initial data. Our method is based on an asymptotic inequality satisfied by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>u</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>T</m:mi> <m:mi>e</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {u(Tₑ)} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>T</m:mi> <m:mi>e</m:mi> </m:msub> </m:math> {Tₑ} that leads to an asymptotic stability estimate for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>T</m:mi> <m:mi>e</m:mi> </m:msub> </m:math> {Tₑ} , allowing us to recover it as the global minimizer of some function. We finally validate our results numerically in two and three dimensions.