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Identification of space-dependent coefficients in two competing terms of a nonlinear subdiffusion equation

Barbara Kaltenbacher, William Rundell

TL;DR

The paper addresses identifying space-dependent coefficients $p(x)$ and $q(x)$ in a nonlinear subdiffusion model with fractional time derivative $\partial_t^\alpha$ from interior measurements. It introduces a fixed-point reconstruction scheme that yields a pointwise update in space by eliminating $p$ and $q$ using observed data, and proves contraction and local uniqueness under long-time observations and decay to a steady state. Numerical experiments demonstrate feasibility, showing that two-run data outperform two-time data and that stronger nonlinearities $f$ enhance identifiability, with robustness to noise. The results rely on regularity and growth conditions for $f$, and are applicable to models including Fisher–KPP, Allen–Cahn, and related reaction–diffusion systems, highlighting practical guidance on observation strategies for coefficient recovery in subdiffusive processes.

Abstract

We consider a (sub)diffusion equation with a nonlinearity of the form $pf(u)-qu$, where $p$ and $q$ are space dependent functions. Prominent examples are the Fisher-KPP, the Frank-Kamenetskii-Zeldovich and the Allen-Cahn equations. We devise a fixed point scheme for reconstructing the spatially varying coefficients from interior observations a) at final time under two different excitations b) at two different time instances under a single excitation. Convergence of the scheme as well as local uniqueness of these coefficients is proven. Numerical experiments illustrate the performance of the reconstruction scheme.

Identification of space-dependent coefficients in two competing terms of a nonlinear subdiffusion equation

TL;DR

The paper addresses identifying space-dependent coefficients and in a nonlinear subdiffusion model with fractional time derivative from interior measurements. It introduces a fixed-point reconstruction scheme that yields a pointwise update in space by eliminating and using observed data, and proves contraction and local uniqueness under long-time observations and decay to a steady state. Numerical experiments demonstrate feasibility, showing that two-run data outperform two-time data and that stronger nonlinearities enhance identifiability, with robustness to noise. The results rely on regularity and growth conditions for , and are applicable to models including Fisher–KPP, Allen–Cahn, and related reaction–diffusion systems, highlighting practical guidance on observation strategies for coefficient recovery in subdiffusive processes.

Abstract

We consider a (sub)diffusion equation with a nonlinearity of the form , where and are space dependent functions. Prominent examples are the Fisher-KPP, the Frank-Kamenetskii-Zeldovich and the Allen-Cahn equations. We devise a fixed point scheme for reconstructing the spatially varying coefficients from interior observations a) at final time under two different excitations b) at two different time instances under a single excitation. Convergence of the scheme as well as local uniqueness of these coefficients is proven. Numerical experiments illustrate the performance of the reconstruction scheme.
Paper Structure (7 sections, 6 theorems, 79 equations, 2 tables, 1 algorithm)

This paper contains 7 sections, 6 theorems, 79 equations, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. The model
  3. The inverse problem
  4. Fixed point schemes
  5. Local uniqueness and convergence of the fixed point scheme
  6. Reconstruction results
  7. Possible alternative observation scenarios

Key Result

Theorem 3.1

For $u_0\in H^1(\Omega)$, $f\in C^2(\mathbb{R})$, under conditions f0fprime0ellipticityL, ellipticregularityL, pinfty, growth, cond_r_kappa2_hi, $r\in L^\infty(0,\infty;L^2(\Omega))$, decay_r, bndy_const, there exist $\rho_0>0$, $C>0$, $0<c_0<c_1$ independent of $T$ such that $u^\sim=u-u^\infty$ (wh as well as the decay estimate decay_usim provided In case $T=\infty$, we also have Here $\eta(t)

Theorems & Definitions (8)

  • Theorem 3.1: Theorem 2.1 in frac_pq_decay
  • Theorem 3.2: Theorem 2.2 in frac_pq_decay
  • Theorem 3.3: Theorem 2.3 in frac_pq_decay
  • Theorem 3.4
  • Theorem 3.5
  • proof
  • Theorem 3.6
  • Remark 3.1