A unique continuation property for a class of parabolic differential inequalities in a bounded domain

Guojie Zheng; Dihong Xu; Taige Wang

doi:10.3934/cpaa.2020280

Article Contents

2021, Volume 20, Issue 2: 547-558. Doi: 10.3934/cpaa.2020280

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A unique continuation property for a class of parabolic differential inequalities in a bounded domain

1.
College of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, China
2.
College of Engineering, Huazhong Agricultural University, Wuhan, 430070, China
3.
Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221, USA

^* Corresponding author
^* Corresponding author

Received: April 30, 2020

Revised: August 31, 2020

Early access: December 2020

Published: February 2021

Guojie Zheng is supported by the Natural Science Foundation of Henan Province (No. 202300410248) and the Natural Science Foundation of Henan Province (No. 2019PL15), and Taige Wang is supported by Faculty Development Fund granted by McMicken College of Arts and Sciences, University of Cincinnati

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Abstract

This article is concerned with a strong unique continuation property of a forward differential inequality abstracted from parabolic equations proposed on a convex domain $ \Omega $ prescribed with some regularity and growth conditions. Our results show that the value of the solutions can be determined uniquely by its value on an arbitrary open subset $ \omega $ in $ \Omega $ at any given positive time $ T $. We also derive the quantitative nature of this unique continuation, that is, the estimate of a $ L^2(\Omega) $ norm of the initial data, which is majorized by that of solution on the bounded open subset $ \omega $ at terminal moment $ t = T $.

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Mathematics Subject Classification: Primary: 35K05, 93B07, 93D15.

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