An Inverse Almost Periodic Problem for a Semilinear Strongly Damped Wave Equation

TL;DR

This paper addresses an inverse boundary value problem for a semilinear strongly damped wave equation, establishing existence, uniqueness, and periodicity of solutions with overdetermination conditions.

Contribution

It introduces a method to determine a time-dependent source coefficient and extends solutions from finite intervals to the entire real line for periodic and almost periodic data.

Findings

Existence and uniqueness of solutions on finite intervals

Extension of solutions to the whole real line

Periodic and almost periodic solutions for periodic data

Abstract

This paper investigates an inverse boundary value problem for a semilinear strongly damped wave equation with Dirichlet boundary conditions in Sobolev spaces of functions bounded in time on $\R$, including periodic and almost periodic functions. In addition to constructing a bounded strong solution, we determine a time-dependent source coefficient via an integral overdetermination condition ensuring well-posedness. After reducing the inverse problem to a direct one, we first establish existence and uniqueness of solutions to an associated problem on finite time intervals. We then extend these solutions to half-lines and construct a bounded strong solution on the whole real line as a limit of such extensions, and subsequently establish its uniqueness. In particular, periodic and almost periodic data yield periodic and almost periodic solutions.