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

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.

💡 Research Summary

The paper addresses the inverse problem of simultaneously identifying two spatially varying coefficients, p(x) and q(x), in a nonlinear subdiffusion (or diffusion) equation of the form

 ∂ₜ^α u – ∇·(D∇u) = q(x) u – p(x) f(u),  (1)

where ∂ₜ^α denotes the Caputo fractional derivative with order 0 < α ≤ 1, D > 0 is a smooth diffusion tensor, and f ∈ C²(ℝ) is a known nonlinearity satisfying f(0)=f′(0)=0. Classical models such as Fisher‑KPP (f(u)=u²), Frank‑Kamenetskii‑Zeldovich (f(u)=u³) and Allen‑Cahn are special cases.

Because there are two unknown spatial functions, at least two independent measurements are required. The authors propose two practical measurement scenarios:

1. Two‑run experiment – Apply two distinct source terms r₁(t,x) and r₂(t,x) to the system, let the solution evolve to a common final time T, and record the interior states g₁(x)=u₁(T,x) and g₂(x)=u₂(T,x).

2. Two‑time experiment – Use a single source r(t,x) but record the interior state at two different times T₁ and T₂, yielding g₁(x)=u(T₁,x) and g₂(x)=u(T₂,x).

Both setups rely only on interior observations, which are often the only data available in applications (e.g., imaging, sensor networks).

The core of the reconstruction method is a fixed‑point iteration. By substituting the PDE into the residuals

 res_i = ∂ₜ^α u_i(T) – r_i(T) – L g_i,  i = 1,2,

and eliminating the unknowns algebraically, the authors obtain a pointwise update formula

 (p,q) = T(p,q) := (1/ det g)