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Inverse source problems with reduced interior data for a coupled reaction-diffusion system

Xinyue Luo, Masahiro Yamamoto, Jin Cheng

Abstract

We consider a two-component semilinear reaction-diffusion system in a bounded spatial domain $Ω$ over a time interval $(0,T)$, which governs the water density $u(x,t)$ and the vegetation biomass density $v(x,t)$ for $x\inΩ$ and $0<t<T$. In this system, called the Klausmeier-Gray-Scott model, we assume that an unknown source depends only on the spatial variable and appears in the reaction-diffusion equation for $u$. The main subject is the inverse source problem of determining a source term from limited data on $(u,v)$. We establish two kinds of stability estimates by means of Carleman estimates. First, a Carleman estimate with a singular weight yields a Lipschitz stability estimate for the inverse source problem from data consisting of a snapshot $u(\cdot,t_0)$ in $Ω$ and $(u,v)$ in a subdomain $ω$ over a time interval. Second, without assuming boundary data, we prove a Hölder stability estimate in any interior subdomain $Ω_0$ satisfying $\overline{Ω_0}\subsetΩ$. We further study how much the observation data can be reduced while preserving uniqueness and stability in the inverse problem under suitable additional conditions.

Inverse source problems with reduced interior data for a coupled reaction-diffusion system

Abstract

We consider a two-component semilinear reaction-diffusion system in a bounded spatial domain over a time interval , which governs the water density and the vegetation biomass density for and . In this system, called the Klausmeier-Gray-Scott model, we assume that an unknown source depends only on the spatial variable and appears in the reaction-diffusion equation for . The main subject is the inverse source problem of determining a source term from limited data on . We establish two kinds of stability estimates by means of Carleman estimates. First, a Carleman estimate with a singular weight yields a Lipschitz stability estimate for the inverse source problem from data consisting of a snapshot in and in a subdomain over a time interval. Second, without assuming boundary data, we prove a Hölder stability estimate in any interior subdomain satisfying . We further study how much the observation data can be reduced while preserving uniqueness and stability in the inverse problem under suitable additional conditions.

Paper Structure

This paper contains 6 sections, 12 theorems, 108 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Proof of Theorem \ref{['thm:reference']}
  4. Proof of Theorem \ref{['thm:holder']}
  5. Proofs of Corollaries \ref{['cor:red2']}--\ref{['cor:red4']}
  6. Conclusion

Key Result

Theorem 2.1

We arbitrarily choose a subdomain $\omega$ satisfying $\overline{\omega} \subset\Omega$ and a time interval $I:= (t_1, t_2)$ with $0<t_1<t_0<t_2<T$. Let $(u,v,f)$ and $(\widetilde{u},\widetilde{v},\widetilde{f})$ belong to $\mathcal{U}(M)$ and solve eq:forward--eq:bc with the same coefficients and b

Figures (3)

  • Figure 1: Domain geometry and measurements.
  • Figure 2: Carleman weight functions for the singular-weight estimate.
  • Figure 3: The regular weight $\psi(x,t)=e^{\lambda(d(x)-\beta(t-t_0)^2)}$ at three representative locations.

Theorems & Definitions (21)

  • Definition 2.1: Admissible class of solutions $(u,v)$
  • Theorem 2.1: Lipschitz stability with data $(\mathcal{M}_1)$
  • Corollary 2.1: Uniqueness via $\mathcal{M}_2$: observation of $v$ only
  • Corollary 2.2: Uniqueness via $\mathcal{M}_3$: observation of $u$ with $f|_\omega$ given
  • Corollary 2.3: Uniqueness via $\mathcal{M}_4$: two snapshots with observation of $u$
  • Theorem 2.2: Hölder stability on interior subdomains
  • Remark 2.1
  • Corollary 2.4: Uniqueness from a local interior observation
  • Lemma 3.1: Construction of a weight function
  • Lemma 3.2: Estimate on $\partial_t\alpha$
  • ...and 11 more