The local backward heat problem

TL;DR

This paper investigates the local backward heat problem with time-dependent coefficients, proposing a method to recover initial data from partial observations by linking controllability and optimal filtering techniques.

Contribution

It introduces a novel approach combining optimal filtering and controllability to solve the local backward heat problem with time-dependent coefficients.

Findings

Successfully recovers initial data from subdomain observations

Establishes a connection between local and global backward problems

Provides a framework for solving inverse heat problems with partial data

Abstract

In this paper, we study the local backward problem of a linear heat equation with time-dependent coefficients under the Dirichlet boundary condition. Precisely, we recover the initial data from the observation on a subdomain at some later time. Thanks to the "optimal filtering" method of Seidman, we can solve the global backward problem, which determines the solution at initial time from the known data on the whole domain. Then, by using a result of controllability at one point of time, we can connect local and global backward problem.