Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Direct and inverse problem for bi-wave equation with time-dependent coefficients from partial data

Sombuddha Bhattacharyya, Pranav Kumar

TL;DR

We study direct and inverse problems for the perturbed bi-wave operator $\mathscr{L}_{A,B,C,q}=\Box^{2}+A(t,x)\Box+B(t,x)\partial_t+C(t,x)\cdot\nabla_x+q(t,x)$ on a bounded space-time domain with time-dependent coefficients. The approach combines interior and boundary Carleman estimates with the construction of geometric optics solutions for $\mathscr{L}_{A,B,C,q}$ and its adjoint, yielding Light Ray Transform identities that connect coefficient differences to boundary measurements. We prove well-posedness of the direct problem and uniqueness for the inverse problem from a partial input-output operator on the boundary, under natural regularity and compatibility conditions. This work extends inverse problems for hyperbolic PDEs to a higher-order, time-dependent bi-wave operator and provides a framework for recovering lower-order perturbations from partial data, signaling potential for more general bi-wave-type models.

Abstract

In this article, we study a direct and an inverse problem for the bi-wave operator $(\Box^2)$ along with second and lower order time-dependent perturbations. In the direct problem, we prove that the operator is well-posed, given initial and boundary data in suitable function spaces. In the inverse problem, we prove uniqueness of the lower order time-dependent perturbations from the partial input-output operator. The restriction in the measurements are considered by restricting some of the Neumann data over a portion of the lateral boundary.

Direct and inverse problem for bi-wave equation with time-dependent coefficients from partial data

TL;DR

We study direct and inverse problems for the perturbed bi-wave operator on a bounded space-time domain with time-dependent coefficients. The approach combines interior and boundary Carleman estimates with the construction of geometric optics solutions for and its adjoint, yielding Light Ray Transform identities that connect coefficient differences to boundary measurements. We prove well-posedness of the direct problem and uniqueness for the inverse problem from a partial input-output operator on the boundary, under natural regularity and compatibility conditions. This work extends inverse problems for hyperbolic PDEs to a higher-order, time-dependent bi-wave operator and provides a framework for recovering lower-order perturbations from partial data, signaling potential for more general bi-wave-type models.

Abstract

In this article, we study a direct and an inverse problem for the bi-wave operator along with second and lower order time-dependent perturbations. In the direct problem, we prove that the operator is well-posed, given initial and boundary data in suitable function spaces. In the inverse problem, we prove uniqueness of the lower order time-dependent perturbations from the partial input-output operator. The restriction in the measurements are considered by restricting some of the Neumann data over a portion of the lateral boundary.

Paper Structure

This paper contains 7 sections, 10 theorems, 103 equations.

Table of Contents

  1. Introduction and Statement of Results
  2. The Direct Problem
  3. Construction of Geometric optics solutions
  4. Proof of main Theorem
  5. Boundary Carleman estimate
  6. Integral identity involving the coefficients
  7. Recovery of the coefficients

Key Result

Theorem 1.1

Let $A\in W^{3,\infty}(Q)$, $B, C \in W^{2,\infty}(Q)$ and $q\in L^{\infty}(Q)$ be time-dependent coefficients. Assume $\psi_{j} \in H^{4-j}(\Omega)$ for $j=0,1,2,3$, $f\in H^{4}(\Sigma)$ and $g\in H^{2}(\Sigma)$ satisfying the compatibility conditions Then there exists a unique solution $u(t)$ of IBVP such that and a constant $C>0$ such that

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • proof
  • Corollary 2.1
  • proof : Proof of the theorem \ref{['IBVP-Biwave-wellposedness']}
  • Proposition 3.1: Interior Carleman estimate
  • proof
  • Proposition 3.2
  • Proposition 3.3
  • ...and 9 more