Observability inequalities for the heat equation with bounded potentials on the whole space

TL;DR

This paper proves an observability inequality for the heat equation with bounded potentials on the entire space, enabling control of the total energy through localized measurements, and demonstrates null controllability.

Contribution

It establishes a new observability inequality for the heat equation on the whole space with bounded potentials, using the parabolic frequency function method.

Findings

Null controllability holds for the heat equation with bounded potentials on the whole space.

The observability inequality links total energy to localized energy in a subdomain.

Method adapted from parabolic frequency function technique.

Abstract

In this paper we establish an observability inequality for the heat equation with bounded potentials on the whole space. Roughly speaking, such a kind of inequality says that the total energy of solutions can be controlled by the energy localized in a subdomain, which is equidistributed over the whole space. The proof of this inequality is mainly adapted from the parabolic frequency function method, which plays an important role in proving the unique continuation property for solutions of parabolic equations. As an immediate application, we show that the null controllability holds for the heat equation with bounded potentials on the whole space.