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An inverse problem for semilinear wave equations on metric tree graphs

Sergei Avdonin, Matti Lassas, Jinpeng Lu, Medet Nursultanov, Lauri Oksanen

Abstract

We study the inverse problem for a semilinear wave equation on metric tree graphs. From the Dirichlet-to-Neumann map defined at all but one of the boundary vertices, we recover unknown connectivity of the graph, lengths of the edges, the time-independent potential and the time-dependent coefficient of the nonlinear term of the equation.

An inverse problem for semilinear wave equations on metric tree graphs

Abstract

We study the inverse problem for a semilinear wave equation on metric tree graphs. From the Dirichlet-to-Neumann map defined at all but one of the boundary vertices, we recover unknown connectivity of the graph, lengths of the edges, the time-independent potential and the time-dependent coefficient of the nonlinear term of the equation.

Paper Structure

This paper contains 12 sections, 15 theorems, 129 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Inverse problems for linear and nonlinear equations on metric graphs.
  3. Main result
  4. Preliminaries and inverse problems for linear wave equation
  5. Metric graphs and Hilbert spaces on graphs
  6. Inverse dynamical and spectral problems for the IBVP \ref{['eq-wave-linear-KN']}
  7. Geometric optics on metric graphs
  8. Geometric optics on a star graph
  9. Linearization method
  10. Inverse problem for trees
  11. Direct problem
  12. Acknowledgments

Key Result

Theorem 1

Let $\Omega$ be a finite metric tree and $\Gamma$ be the set of all leaves of $\Omega$. Let $\gamma_0$ be one leaf and denote $\Gamma_0=\Gamma\setminus \{\gamma_0\}.$ Suppose that we are given the Dirichlet-to-Neumann map $\Lambda_T$ on $\Gamma_0$ for the semilinear wave equation eq-wave-nonlinear u where $d$ is the distance function on the metric tree. Assume $T>2D(\gamma_0)$. Then $\Lambda_T$ un

Figures (8)

  • Figure 1: As a special case of Theorem \ref{['main-tree']}, the reconstruction of the time-dependent coefficient $a$ on the interval $[0,L]$. Suppose control and observation of waves are available at only one endpoint $x=0$, i.e., $\Gamma_0=\{0\}$ and $\gamma_0=\{1\}$. Then $\Lambda_T$ uniquely determines $a$ on the domain \ref{['Domain_recovery']}, the one enclosed by the blue lines.
  • Figure 2: Setting of Lemma \ref{['interval']} and \ref{['star-equal']}.
  • Figure 3: Setting of Proposition \ref{['star-diff']}.
  • Figure 4: The support of the wave $v_1$ (translated so that the wave starts from the origin).
  • Figure 5: A two-step procedure for reconstructing $a$ on $[l_1,L]$. The blue lines represent the support of $v_2,v_3$.
  • ...and 3 more figures

Theorems & Definitions (29)

  • Theorem 1
  • Remark 1.1
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 4.1
  • proof
  • ...and 19 more