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Inverse problem for wave equation of memory type with acoustic boundary conditions: Global solvability

Zhanna D. Totieva, Kush Kinra, Manil T. Mohan

TL;DR

The paper studies the one-dimensional inverse problem of identifying the memory kernel in a memory-type wave equation with acoustic boundary conditions, using an integral overdetermination data. By reformulating the problem as an equivalent system for $(v,k,y)$ with homogeneous boundaries, the authors employ contraction mappings in Sobolev spaces and energy estimates to prove global existence and uniqueness of the solution. The analysis hinges on precise convolution inequalities and a careful fixed-point argument that is first local-in-time and then extended globally. These results advance kernel identification in viscoelastic/memory mediums and provide a rigorous framework for global solvability in this class of hyperbolic integro-differential problems.

Abstract

In this article, we study the one-dimensional inverse problem of determining the memory kernel by the integral overdetermination condition for the direct problem of finding the velocity potential and the displacement of boundary points. A wave equation with initial and acoustic boundary conditions in media with dispersion is used as a mathematical model. The inverse problem is reduced to an equivalent problem with homogeneous boundary conditions for the system of integro-differential equations. Using the technique of estimating integral equations and the contraction mappings principle in Sobolev spaces, the global existence and uniqueness theorem for the inverse problem is proved.

Inverse problem for wave equation of memory type with acoustic boundary conditions: Global solvability

TL;DR

The paper studies the one-dimensional inverse problem of identifying the memory kernel in a memory-type wave equation with acoustic boundary conditions, using an integral overdetermination data. By reformulating the problem as an equivalent system for with homogeneous boundaries, the authors employ contraction mappings in Sobolev spaces and energy estimates to prove global existence and uniqueness of the solution. The analysis hinges on precise convolution inequalities and a careful fixed-point argument that is first local-in-time and then extended globally. These results advance kernel identification in viscoelastic/memory mediums and provide a rigorous framework for global solvability in this class of hyperbolic integro-differential problems.

Abstract

In this article, we study the one-dimensional inverse problem of determining the memory kernel by the integral overdetermination condition for the direct problem of finding the velocity potential and the displacement of boundary points. A wave equation with initial and acoustic boundary conditions in media with dispersion is used as a mathematical model. The inverse problem is reduced to an equivalent problem with homogeneous boundary conditions for the system of integro-differential equations. Using the technique of estimating integral equations and the contraction mappings principle in Sobolev spaces, the global existence and uniqueness theorem for the inverse problem is proved.
Paper Structure (8 sections, 10 theorems, 141 equations)

This paper contains 8 sections, 10 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. The concrete version of the inverse problem
  3. Function spaces
  4. The inverse problem
  5. Important inequalities and results
  6. The Equivalent Problem
  7. The Main Result
  8. Energy estimates for a linear problem

Key Result

Lemma 2.3

Let $X$ be a Banach space, $p \in (1, \infty),\ \tau\in \mathbb{R}_+$, $k\in L^p(0,\tau)$, and $f\in L^p(0,\tau; X)$. Then $k \ast f \in L^p(0, \tau ; X)$ and where $(k \ast f)(t) = \int_0^t k(t-s)f(s)ds$.

Theorems & Definitions (21)

  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3: Colombo+Guidetti_2007
  • proof
  • Lemma 2.4: Colombo+Guidetti_2007
  • Lemma 3.1
  • proof
  • Theorem 4.1: Local-in-time existence
  • proof : Proof of Theorem \ref{['thm-4.1']}
  • Lemma 4.2
  • ...and 11 more