[Paper Review] Unique determination of coefficients and kernel in nonlocal porous medium equations with absorption term

Yi‐Hsuan Lin, Philipp Zimmermann|arXiv (Cornell University)|May 25, 2023

Numerical methods in inverse problems4 citations

TL;DR

This paper establishes the unique determination of the coefficients $\rho$, $q$, and kernel $K$ in a nonlocal porous medium equation with absorption term $\rho\partial_t u + L_K(u^m) + qu = 0$ ($m>1$) from the partial Dirichlet-to-Neumann map. Using a Galerkin method for existence, a comparison principle, and asymptotic analysis under the unique continuation property, it proves that the DN map uniquely identifies $\rho$, $K$, and $q$ under uniform ellipticity and regularity assumptions, including for the fractional conductivity operator.

ABSTRACT

The main purpose of this article is the study of an inverse problem for nonlocal porous medium equations (NPMEs) with a linear absorption term. More concretely, we show that under certain assumptions on the time-independent coefficients $ρ,q$ and the time-independent kernel $K$ of the nonlocal operator $L_K$, the (partial) Dirichlet-to-Neumann map uniquely determines the three quantities $(ρ,K,q)$ in the nonlocal porous medium equation $ρ\partial_tu+L_K(u^m)+qu=0$, where $m>1$. In the first part of this work we adapt the Galerkin method to prove existence and uniqueness of nonnegative, bounded solutions to the homogenoeus NPME with regular initial and exterior conditions. Additionally, a comparison principle for solutions of the NPME is proved, whenever they can be approximated by sufficiently regular functions like the one constructed for the homogeneous NPME. These results are then used in the second part to prove the unique determination of the coefficients $(ρ,K,q)$ in the inverse problem. Finally, we show that the assumptions on the nonlocal operator $L_K$ in our main theorem are satisfied by the fractional conductivity operator $\mathcal{L}_γ$, whose kernel is $γ^{1/2}(x)γ^{1/2}(y)/|x-y|^{n+2s}$ up to a normalization constant.

Motivation & Objective

• To address the inverse problem of identifying unknown coefficients and nonlocal kernel in a quasilinear nonlocal PDE with absorption.
• To establish uniqueness of the coefficients $\rho$, $K$, and $q$ in the nonlocal porous medium equation from partial boundary measurements.
• To extend inverse problem techniques from linear nonlocal operators to quasilinear equations with absorption.
• To verify that the assumptions on the nonlocal operator are satisfied by the fractional conductivity operator $\mathcal{L}_\gamma$.

Proposed method

• Adapts the Galerkin method to prove existence and uniqueness of nonnegative, bounded solutions to the homogeneous nonlocal porous medium equation with regular initial and exterior data.
• Establishes a comparison principle for solutions under approximation by sufficiently regular functions.
• Introduces an integral-time transformation to analyze the asymptotic behavior of solutions near the initial time.
• Applies asymptotic analysis to extract information about the coefficients from the Dirichlet-to-Neumann map.
• Uses the unique continuation principle and Runge approximation properties to link the DN map to the unknown coefficients.
• Verifies that the fractional conductivity operator $\mathcal{L}_\gamma$, with kernel $K(x,y) = C_{n,s} \gamma^{1/2}(x)\gamma^{1/2}(y)$, satisfies the required assumptions on $K$.

Experimental results

Research questions

• RQ1 Can the coefficients $\rho$, $q$, and the kernel $K$ in a nonlocal porous medium equation with absorption term be uniquely determined from the partial Dirichlet-to-Neumann map?
• RQ2 Does the comparison principle hold for solutions of the nonlocal porous medium equation when approximated by regular functions?
• RQ3 Is the unique continuation principle applicable to the nonlocal operator $L_K$ in the context of this inverse problem?
• RQ4 Do the assumptions on $K$ hold for the fractional conductivity operator $\mathcal{L}_\gamma$?
• RQ5 Can the DN map uniquely identify $\rho$, $K$, and $q$ under the given regularity and ellipticity conditions?

Key findings

• The partial Dirichlet-to-Neumann map uniquely determines the time-independent coefficients $\rho$, $q$, and the kernel $K$ of the nonlocal operator $L_K$ in the nonlocal porous medium equation with absorption.
• Existence and uniqueness of nonnegative, bounded solutions to the homogeneous nonlocal porous medium equation are established via the Galerkin method under regular initial and exterior data.
• A comparison principle is proven for solutions that can be approximated by sufficiently regular functions, enabling comparison of sub- and super-solutions.
• The asymptotic analysis of the transformed solution reveals that the DN map encodes information about $\rho$, $K$, and $q$ in the limit as time approaches zero.
• The assumptions on the nonlocal operator $L_K$ are satisfied by the fractional conductivity operator $\mathcal{L}_\gamma$, whose kernel is $K(x,y) = C_{n,s} \gamma^{1/2}(x)\gamma^{1/2}(y)$ for uniformly elliptic $\gamma$.
• The unique continuation principle is used as a foundational tool to ensure that the DN map determines the full coefficient structure, particularly in the absence of additional symmetry assumptions.