The property of unique continuation for second order evolution PDEs

TL;DR

This paper introduces a straightforward method using two-parameter Carleman inequalities to establish the unique continuation property for classical second-order evolution PDEs like wave, parabolic, and Schrödinger equations in cylindrical domains.

Contribution

It provides a simple, self-contained approach to prove unique continuation for second-order evolution PDEs, extending classical results with some modifications.

Findings

Established unique continuation for wave, parabolic, and Schrödinger operators.

Utilized two-parameter Carleman inequalities with pseudo-convex hypersurfaces.

Reproduced and slightly modified classical results.

Abstract

We present a simple and self-contained approach to establish the unique continuation property for some classical evolution equations of second order in a cylindrical domain. We namely discuss this property for wave, parabolic and Sch\"odinger operators with time-independent principal part. Our method is builds on two-parameter Carleman inequalities combined with unique continuation across a pseudo-convex hypersurface with respect to the space variable. The most results we demonstrate in this work are more or less classical. Some of them are not stated exactly as in their original form.