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Well-posedness and inverse problems for semilinear nonlocal wave equations

Yi-Hsuan Lin, Teemu Tyni, Philipp Zimmermann

Abstract

This article is devoted to forward and inverse problems associated with time-independent semilinear nonlocal wave equations. We first establish comprehensive well-posedness results for some semilinear nonlocal wave equations. The main challenge is due to the low regularity of the solutions of linear nonlocal wave equations. We then turn to an inverse problem of recovering the nonlinearity of the equation. More precisely, we show that the exterior Dirichlet-to-Neumann map uniquely determines homogeneous nonlinearities of the form $f(x,u)$ under certain growth conditions. On the other hand, we also prove that initial data can be determined by using passive measurements under certain nonlinearity conditions. The main tools used for the inverse problem are the unique continuation principle of the fractional Laplacian and a Runge approximation property. The results hold for any spatial dimension $n\in \N$.

Well-posedness and inverse problems for semilinear nonlocal wave equations

Abstract

This article is devoted to forward and inverse problems associated with time-independent semilinear nonlocal wave equations. We first establish comprehensive well-posedness results for some semilinear nonlocal wave equations. The main challenge is due to the low regularity of the solutions of linear nonlocal wave equations. We then turn to an inverse problem of recovering the nonlinearity of the equation. More precisely, we show that the exterior Dirichlet-to-Neumann map uniquely determines homogeneous nonlinearities of the form under certain growth conditions. On the other hand, we also prove that initial data can be determined by using passive measurements under certain nonlinearity conditions. The main tools used for the inverse problem are the unique continuation principle of the fractional Laplacian and a Runge approximation property. The results hold for any spatial dimension .
Paper Structure (13 sections, 11 theorems, 138 equations)

This paper contains 13 sections, 11 theorems, 138 equations.

Table of Contents

  1. Introduction
  2. Mathematical model and main results
  3. Earlier literature
  4. Organization of the paper
  5. Preliminaries
  6. Fractional Sobolev spaces and fractional Laplacian
  7. Bochner spaces
  8. The forward problem of nonlinear nonlocal wave equations
  9. Well-posedness
  10. The DN map
  11. Inverse problems for the nonlinear nonlocal wave equation
  12. Runge approximation
  13. Proofs related to the inverse problems

Key Result

Theorem 1.1

Let $\Omega \subset {\mathbb R}^n$ be a bounded domain with Lipschitz boundary, $T>0$ and $s>0$ a non-integer. Let $W_1,W_2\subset \Omega_e$ be open sets. Suppose the nonlinearities $f_j$ satisfy the conditions in Assumption main assumptions on nonlinearities with $F(x,\tau)\geq 0$. Suppose also tha for $j=1,2$, satisfying for any $\varphi \in C^\infty_c((W_1)_T)$. Then there holds $f_1(x,\tau)=f

Theorems & Definitions (29)

  • Theorem 1.1: Recovery of the nonlinearity
  • Theorem 1.2: Recovery of the initial values by passive measurements
  • Remark 1.3
  • Corollary 1.4: Simultaneous recovery of both initial data and coefficients
  • Proposition 2.1: UCP for fractional Laplacians
  • Proposition 2.2: Poincaré inequality (cf. RZ-unbounded)
  • Lemma 2.3
  • Theorem 3.1: Well-posedness of linear nonlocal wave equation
  • Remark 3.2
  • proof
  • ...and 19 more