Unique continuation and observability for a semi-linear parabolic system

¹ College of Digital Technology and Engineering Ningbo University of Finance & Economics, Ningbo 315175, PR China
² Department of Mathematical Sciences University of Cincinnati, Cincinnati, OH 45221, USA

^* Corresponding author: guojiezheng@yeah.net

Received: 22 August 2023
Accepted: 27 February 2024

Abstract

This article is concerned with the unique continuation and observability of a semi-linear parabolic system in a bounded domain Ω with homogeneous Dirichlet boundary conditions. We first study the existence and uniqueness of the L^∞-solution to the system when the initial value belongs to L^∞(Ω)². We then build up some observation estimates for this system. As a consequence, we can conclude if two solutions are equal over a nonempty open subset ω ⊂ Ω at some time T > 0, then they coincide over Ω × [0, T].